# Performance and Effectiveness of the Passive-Compliant Citrus-Picking Manipulator
## 3 Parameter Optimization of Picking Manipulator
### Grasping Component Force Analysis and Component Selection  
#### 3.3.1 Force Analysis of Grasping Member  
Since the four gripping components are evenly distributed on the surface of the citrus, force analysis is performed on one side, as shown in Figure 6. Point \(O\) is the pivot point of the gripping component; \(N_{1}^{\prime}\) is the normal force at the first contact point, in N; \(N_{2}^{\prime}\) is the normal force at the second contact point, in N; \(N_{a}\) is the resultant force of \(N_{1}^{\prime}\) and \(N_{2}^{\prime}\), in N; \(l_{5}\) is the perpendicular distance from the pivot point \(O\) to the sliding block movement surface, in mm; \(l_{6}\) is the distance from the pivot point \(O\) to the proximal end of the middle finger, in mm; \(S\) is the stroke of the sliding block, in mm; \(F^{\prime}\) and \(F^{\prime\prime}\) are the components of force \(F\) in two directions, in N; \(\delta\) is the rotational angle of the gripper relative to the pivot point \(O\), in \({}^{\circ}\); \(\theta\) is the angle between the distal and middle phalanges, in \({}^{\circ}\). The distance from point \(O\) to the spring force \(F\) is \(l_{5}=31.3\) mm, and the distance from point \(O\) to the proximal end of the middle finger segment is \(l_{6}=38\) mm. \(N_{1}^{\prime}\) and \(N_{2}^{\prime}\) represent the normal forces acting on the gripping components, and their resultant force is \(N_{a}\), with the action line passing through point a. Since \(N_{1}^{\prime}\) and \(N_{2}^{\prime}\) are equal in magnitude and symmetric about point \(a\), the resultant moment \(M_{a}\) is 0, meaning that point a only experiences the resultant force \(N_{a}\). Assuming the gripping component is an ideal mechanism (i.e., ignoring friction), the forces on the gripping component are simplified to point \(O\). Based on the force moment equilibrium equation, as shown in Equation (10), the required spring force \(F\) is derived.  
\[\sum_{i=1}^{n}M_{i}=M_{q}+M_{p}=0 \tag{10}\]  
In the equation, n = 2; \(M_{q}\) is the resultant moment of the force \(N_{a}\) about point \(O\), in N\(\cdot\)m; \(M_{p}\) is the resultant moment of the spring force \(F\) about point \(O\), in N\(\cdot\)m. The equations are as follows:  
\[M_{q}=N_{a}\bigg{(}l_{2}\sin\frac{\theta}{2}+l_{6}\bigg{)} \tag{11}\]  
\[N_{a}=2N_{1}^{\prime}\sin\frac{\theta}{2} \tag{12}\]  
\[M_{p}=-F^{\prime}\frac{l_{5}}{\cos\delta} \tag{13}\]  
\[F=\frac{F^{\prime}}{\cos\delta} \tag{14}\]  
By solving Equations (10) to (14) simultaneously, we obtain the following:  
\[F=\frac{2N_{1}^{\prime}\sin\frac{\theta}{2}\bigg{(}l_{2}\sin\frac{\theta}{2}+ l_{6}\bigg{)}}{l_{5}} \tag{15}\]  
Figure 5: Stress analysis of the citrus XZ plane.  
From Equation (15), it can be seen that \(l_{2}\), \(l_{5}\), \(l_{6}\), and \(\theta\) are inherent parameters of the gripper, and the spring force \(F\) depends only on the normal force \(N_{1}^{\prime}\) exerted on the gripper, since \(N_{1}^{\prime}\) is the reaction force of \(N_{1}\), equal in magnitude and opposite in direction. The citrus fruit bears a safe load of 4 to 9 N, so the required spring force range is from 17.1 to 38.4 N.