# Performance and Effectiveness of the Passive-Compliant Citrus-Picking Manipulator
## 3 Parameter Optimization of Picking Manipulator
### Stress Analysis of Citrus
The longitudinal elastic modulus of the citrus peel and flesh are 11.408 MPa and 0.277 MPa, respectively, with ultimate stresses of 1.250 MPa and 0.048 MPa. When the gripping force is 12 N, the maximum equivalent stresses in the citrus peel and flesh are 1.091 MPa and 0.035 MPa, both of which are lower than their respective ultimate stresses, meaning no mechanical damage will occur to the citrus [24; 33]. When the load on the citrus is less than 10 N, the citrus remains within the elastic range, and the elastic deformation is minimal, with deformation being less than 6 mm. Therefore, it can be considered safe for the citrus when the harvesting manipulator exerts a force of less than 10 N during robotic harvesting [19]. Therefore, the normal force range can be determined as 0.56 to 10 N.  
To ensure stable grasping and prevent slipping of the maximum-sized citrus, the minimum normal force required at each contact point is \(N_{\rm min}\). At this point, \(\theta_{1}\) is 69.38\({}^{\circ}\) and \(\theta_{2}\) is 19.38\({}^{\circ}\), with the primary force analysis shown in Figure 5. \(f_{i}\) represents the static friction force along the finger surface of the \(i\)-th contact point, in N; \(N_{i}\) represents the normal force at the \(i\)-th contact point, in N; \(\theta_{1}\) represents the angles between \(f_{1}\),\(f_{3}\),\(f_{5}\),\(f_{7}\) and the horizontal plane, in \({}^{\circ}\); \(\theta_{2}\) represents the angles between \(N_{2}\), \(N_{4}\), \(N_{6}\), \(N_{8}\) and the horizontal plane, in \({}^{\circ}\); \(mg\) represents the gravitational force of the citrus, in N. The citrus with a maximum diameter, \(D_{\rm max}\), of 100 mm has a mass of approximately 340 g. By increasing the margin coefficient by 10%, the fruit mass is taken as 374 g, and solving Equations (7) to (9) yields a minimum normal force \(N_{\rm min}\) of 0.92 N. Thus, the load borne by the citrus during stable grasping is between 0.92 and 10 N. The contact between the harvesting manipulator and the citrus surface is not a simple point-to-surface contact, but a complex multi-point contact problem. The surface of the citrus has a certain curvature and is not smooth, and its surface characteristics affect the distribution and contact area. Therefore, the contact relationship should be considered as a combination of multiple contact points, rather than simple point or surface contact. Additionally, as citrus is a flexible body, it undergoes local deformation when subjected to force, and the degree of deformation may vary with changes in external forces. Different contact areas experience varying degrees of pressure, leading to differences in the deformation of the peel and the stress distribution in the flesh. Therefore, deformation factors need to be introduced in the force analysis to more accurately describe the mechanism of contact and force transmission. This force analysis does not account for these complex nonlinear behaviors. Future research should consider contact forces, friction forces, and the deformation response of the fruit to more comprehensively simulate the force conditions during the harvesting process. Therefore, this study selects a safe load range from 4 to 9 N.