# Performance and Effectiveness of the Passive-Compliant Citrus-Picking Manipulator
## 3 Parameter Optimization of Picking Manipulator
### Stress Analysis of Citrus  
Due to the 4 V-shaped gripping components having a total of 8 contact points with the citrus, the force on the \(i\)-th contact point mainly consists of the normal force \(N_{i}\), the frictional force\(f_{i}\), and the gravitational force of the citrus \(mg\). When grasping the citrus with a \(D_{\text{min}}\) of 65 mm, the force on the citrus is simulated using CAD 2022 software, resulting in an angle \(\theta_{1}\) of 53.17\({}^{\circ}\)and an angle \(\theta_{2}\) of 3.17\({}^{\circ}\). The main force analysis is shown in Figure 4. \(f_{1}\) represents the static friction force along the gripping surface at the \(i\)-th contact point (N); \(N_{i}\) represents the normal force at the \(i\)-th contact point (N); \(\theta_{1}\) is the angle between \(f_{1}\),\(f_{3}\), \(f_{5}\) and \(f_{7}\) and the horizontal plane (\({}^{\circ}\)); \(\theta_{2}\) is the angle between \(N_{2}\), \(N_{4}\), \(N_{6}\), and \(N_{8}\) and the horizontal plane (\({}^{\circ}\)); \(mg\) is the gravitational force of the citrus fruit (N). The 4 V-shaped gripping components of the harvesting robotic hand are evenly distributed on the surface of the citrus, and the normal force, \(N_{i}\), on each contact point is equal, satisfying Equation (2). The forces in the horizontal direction satisfy Equations (3)-(6), so the resultant force on the citrus in the horizontal direction is 0. To ensure that the smallest size of citrus is stably gripped without slipping, the citrus force balance equation should satisfy Equation (7). Under this condition, when the maximum static friction between the gripper component and the fruit is reached, Equation (8) is satisfied. At this point, the normal force on each contact point is minimized, with the minimum normal force denoted as \(N_{\text{min}}\), satisfying the condition in Equation (9). \[N_{i}=N_{1}=N_{2}=N_{3}\cdots N_{8} \tag{2}\]  
\[f_{1}\cos\theta_{1}-f_{3}\cos\theta_{1}=0 \tag{3}\]  
\[f_{4}\sin\theta_{2}-f_{2}\sin\theta_{2}=0 \tag{4}\]  
\[f_{5}\cos\theta_{1}-f_{7}\cos\theta_{1}=0 \tag{5}\]  
\[f_{8}\sin\theta_{2}-f_{6}\sin\theta_{2}=0 \tag{6}\]  
\[\left( \begin{array}{c}f_{1}+f_{3}+\\ f_{5}+f_{7}\end{array} \right)\sin\theta_{1}+\left( \begin{array}{c}f_{2}+f_{4 }+\\ f_{6}+f_{8}\end{array} \right)\cos\theta_{2}+\left( \begin{array}{c}N_{2}+N_{4 }\\ +N_{6}+N_{8}\end{array} \right)\sin\theta_{2}-\left( \begin{array}{c}N_{1}+N_{ 3}+\\ N_{5}+N_{7}\end{array} \right)\cos\theta_{1}-mg=0 \tag{7}\]  
\[f_{i}=\mu N_{i} \tag{8}\]  
\[N_{\text{min}}\geq N_{i} \tag{9}\]  
Figure 4: Force analysis of citrus and force analysis of citrus XZ plane. In the equation, \(m\) represents the mass of the citrus fruit (kg), \(g\) is the gravitational acceleration (taken as 9.8 m/s\({}^{2}\)), and \(\mu\) is the coefficient of friction between the citrus peel and the cushioning material. Attaching a certain amount of cushioning material to the surface of the V-shaped gripping components can increase the friction coefficient, improve the compliance of the finger surface, reduce the required normal force, and reduce potential mechanical damage during the gripping process. Different cushioning materials have varying friction coefficients with the citrus peel, and there are three common types: rubber, silicone, and pearl cotton. In this study, silicone is chosen as the cushioning material, with \(\mu\) taken as 0.68 [19]. The citrus with a minimum diameter, \(D_{\rm min}\), of 65 mm has a mass of approximately 140 g. By increasing the margin coefficient by 10%, the fruit mass is taken as 154 g, and solving Equations (7) to (9) results in a minimum normal force of 0.56 N. Additionally, because the larger the citrus, the greater the gripping force provided by the harvesting manipulator, the excessive normal force could damage the citrus' biological tissue. Therefore, when considering the citrus with a maximum diameter, \(D_{\rm max}\) of 100 mm, the maximum normal force that the citrus can withstand, denoted as \(N_{\rm max}\), must be considered.