# Design and Implementation of a Multi-Function Gripper for Grasping General Objects
## 3 Grasping Modes and Grasping Strategy
### Parallel Grasping
#### 3.1.2 Analysis of Parallel Grasping Force  
The relationship between the parallel-grasping force and actuation torque can be evaluated with the quasi-static modeling of the finger. As mentioned earlier, during the parallel-grasping sequence, the entire finger moves as a single rigid body through using one motor to control the link \(L_{\text{AD}}\). The parallel grasping model is shown in Figure 9a. By equating the input and output virtual power, we have  
\[\mathbf{\hat{f}}^{T}\cdot\mathbf{v}_{P}=\tau_{a}\omega_{a} \tag{1}\]  
where referring to Figure 9a, represents the contact force at contact point P during parallel grasping, represents the instantaneous velocity of the contact point P, represents the actuation torque exerted on the link used to perform the open-close motion of the finger, represents the angular velocity of the active link, i.e., and denotes the open-close angle of the active link with respect to the horizontal axis.  
During the parallel-grasping sequence, the entire finger moves as a single rigid body to perform the open-close motion. Thus, point K shown in Figure 9a will follow a circular trajectory with its rotation center located at point A. Because the orientation of the distal phalanx remains constant  
Figure 8: Demonstration of parallel grasping.  
Figure 9: (**a**) Parallel-grasping model. (**b**) Trajectory of contact point P.  
during the open-close motion, all its points have the same instantaneous velocity. By defining a virtual parallelogram \(\mathrm{Op}-\mathrm{A}-\mathrm{K}-\mathrm{P}\) (\(\mathrm{O}_{\mathrm{P}}\mathrm{A}~{}=~{}\mathrm{PK}\)) shown in Figure 9a, we know that the contact point P follows a circular trajectory with its rotation center located at the virtual point \(\mathrm{O}_{\mathrm{P}}\).The motion of contact point P during parallel grasping is shown in Figure 9b. The velocity of contact point P is the same as the velocity of point K, which can be derived as  
\[v_{P}=v_{K}=L_{\mathrm{AK}}\dot{\phi} \tag{2}\]  
As we know, the whole finger moves as a single rigid body, we have  
\[\phi+2\pi-\theta_{a}=\theta_{c} \tag{3}\]  
where \(\theta_{c}\) is a constant value that can be found from our design.  
Then we have  
\[\dot{\phi}{=\dot{\theta}_{a}} \tag{4}\]  
From Figure 9a and substituting Equations (2) and (4) into Equation (1), Equation (1) can be rewritten as  
\[\vec{f}^{T}\cdot\mathbf{v}_{P}=fv_{P}\sin\phi=fL_{\mathrm{AK}}\dot{\phi}\sin\phi=fL _{\mathrm{AK}}\dot{\theta}_{a}\sin(\theta_{a}+\theta_{c})=\tau_{a}\dot{\theta} _{a} \tag{5}\]  
Thus, the parallel-grasping force can be obtained as  
\[f=\tau_{a}/(L_{\mathrm{AK}}\sin(\theta_{a}+\theta_{c})) \tag{6}\]  
The power-transmission system consists of the motor gear head and worm-gear set. By considering the power efficiency, the relationship between the actuation torque of independent link \(L_{\mathrm{AD}}\) and input motor torque can be derived as  
\[\tau_{a}=\tau_{m}r_{m}r_{w}\eta_{m}\eta_{w} \tag{7}\]  
where \(\tau_{m}\) is the input motor torque. \(r_{m}\) and \(\eta_{m}\) are the gear ratio and efficiency of the motor gear head, respectively. \(r_{w}\) and \(\eta_{w}\) are the gear ratio and efficiency of the worm-gear set, respectively.  
Moreover, the motor capacity can be determined according to the payload requirement using the above-mentioned relationship between the grasping force and input motor torque.  
Furthermore, because the efficiency coefficients (\(\eta_{m}\) and \(\eta_{w}\)) shown in Equation (7) are difficult to accurately evaluate, accurately measuring the energy efficiency of the power-transmission system becomes difficult. Thus, Equation (6) can only be used to estimate the grasping force. In order to accurately control the grasping force, we need to install a force sensor at the distal phalanx.