*2.1. Finite-Element Analysis of Finger Structure with the Fin Ray Effect*

The harvesting gripper's finger mechanism uses a triangular Fin-Ray soft finger component, which has a passive compliance quality and can implement an envelope while clamping spherical items. The general construction of the finger consists of the front and rear beams, cross beams, and base, as shown in Figure 1a. The front beam comes into contact with the fruits, and the front and rear beams are linked by cross beams. These cross-beam support rods are the foundation of Fin Ray fingers. Because of the presence of these crossbeam support rods, the Fin Ray structure can withstand greater loads than conventional flexible constructions.

**Figure 1.** Characteristics of the Fin Ray finger: (**a**) basic components; (**b**) displacement of the fingertip; (**c**) structure of the Fin Ray finger.

#### 2.1.1. Pre-Preparation of the Simulation Experiment

The finger gripping force must be sufficient to improve the grasping stability. Furthermore, the pressure per unit area of the pericarp should be small enough to guarantee that the fruit pericarp remains intact. As a result, the finger gripping force and the bending degree are two critical criteria. The finger gripping force can ensure clamping stability, while the finger bending degree can assure clamping stability and safety by increasing the contact area between the fingers and the fruits. The stress of the Fin Ray finger during deformation is complicated by making the mathematical modeling difficult. As a result, using the simulated tests, this research investigates the effect of the front and rear beam thickness, the finger width, and the number of cross beams on the finger gripping force and bending degree, as shown in Figure 1a. In the simulation experiment, the contact stress between the finger and apple is used to characterize the gripping force, and the displacement of the fingertip is used to characterize the bending degree, as shown in Figure 1b.

A single finger adopts a symmetrical structure; the total length of the finger is 120 mm, and the front beam and the rear beam are each at an angle of 80◦ to the base. The cross beams are parallel to the base; the distance is equal, and the thickness of the cross beams is 1.40 mm. The little bulges are designed on the cross beams to increase the rigidity and strengthen the load capacity, as shown in Figure 1c.

The TPU 95A [49] was chosen as the finger material. The TPU soft material is a hyperelastic nonlinear material with isotropic properties throughout the stress process. Furthermore, because the bending deformation of the soft finger is a nonlinear large deformation, the Yeoh model can better represent its material properties [50]. The strain energy density function *W* can be written as follows:

$$\mathcal{W} = \sum\_{i=1}^{N} \mathbb{C}\_{i0} (I\_1 - 3)^i + \sum\_{k=1}^{N} \frac{1}{D\_k} (J - 1)^{2k} \,\mathrm{}\,\tag{1}$$

where *N* is the order of the model; *I*<sup>1</sup> is the deformation tensor; *Ci*<sup>0</sup> and *Dk* are the material constants; *J* is the volume ratio. When TPU is regarded as the incompressible material, *J* = 1.

The strain energy density function in the form of the binomial parameters is usually used [51], and the typical binomial parameter form of the Yeoh model is

$$\mathcal{W} = \mathbb{C}\_{10}(I\_1 - \mathfrak{Z}) + \mathbb{C}\_{20}(I\_1 - \mathfrak{Z})^2. \tag{2}$$

The fitting curve of the stress and strain of the TPU 95A was obtained through the uniaxial tensile test, as shown in Figure 2. The material parameters obtained after processing and analysis are shown in Table 1.

**Figure 2.** Strain–stress curve of the tensile test and fitting using the Yeoh model (TPU95A).


**Table 1.** Mechanical property parameters of materials.

Because the contact stress between the three fingers and the fruit is the same, the contact between a single finger and the fruit can be considered to reduce the quantity of simulation calculation, to simplify the analysis.

During the simulation, the center of the bottom plate of the gripper is kept aligned with the center of the fruit at a distance of 65 mm [49].

2.1.2. Influence of Geometric Parameters on Contact Stress and Fingertip Displacement

Each geometric parameter has a varied effect on the contact stress and fingertip displacement. All other parameters were held constant to compare their changes when the given parameters were altered, and the influence of the given single parameter on them was gradually optimized.

First, the influence of the thickness of the front and rear beams was analyzed. The stress increases dramatically as the thickness increases, while the fingertip displacement decreases, as shown in Figure 3a.

**Figure 3.** Changes in stress and displacement according to three factors: (**a**) thickness of front and rear beams; (**b**) width of fingers; (**c**) number of cross beams.

When the thickness of the front and rear beams is 2 mm, the stiffness of the finger after contact with the apple cannot be guaranteed, resulting in a small gripping force and easy fruit slip; when the thickness is 4.5 mm, the stress of the material itself will greatly limit its bending deformation and reduce the contact area between the fingers and fruit. At the same time, because excessive stress might cause fruit damage, the thickness of the front and back beams should not be too tiny or too large. When the thickness is 3.5 mm, the downward trend of the fingertip displacement becomes stronger as the thickness increases, while the upward trend of stress tends to be soft. As a result, selecting a thickness of 3.5 mm for the front and rear beams not only meets the requirement of the increasing gripping force but also allows fingers to make good contact with the fruits.

The effect of the finger width was then investigated. With the increase of the width, the fingertip displacement diminishes. However, the stress does not follow a constant pattern, as shown in Figure 3b. When the width is 10 mm, the stress and fingertip displacement is the greatest. This is because the finger width is excessively narrow, resulting in a limited contact area between the finger and the apple and high contact stress acting on the apple surface, which is easily damaged. Although the degree of the finger bend is greater when the finger is thin, it also results in insufficient grasping stiffness and fruit slide. When the width is 25 mm, the contact area between fingers and fruit increases, but its structure affects its bending, and it is not suitable for collecting fruits in the complex growing environment. When the width is 16 mm, as the width continues to increase, the fingertip displacement decreases dramatically and the stress tends to be flat. As a result, the best finger width is set to 16 mm in this study.

Finally, the number of beams was taken into account. Because the cross beams are the primary components that influence the stiffness of fingers, the number of cross beams has a substantial impact on the Young's modulus of the fingers [25]; hence, the distribution of the cross beams may have a major impact on the gripper performance. In distribution, there are several combinations of the cross beams. For the sake of simplicity, just the simplest equidistant parallel arrangement of the cross beams was considered in this study. Change the thickness of the front and rear beams to 3.5 mm, the width of the fingers to 16 mm, and change the number of cross beams. As the number of cross beams grows, so does the stress, and the fingertip displacement declines first and subsequently increases. When the number of beams is 9, the fingertip displacement reaches the maximum and then decreases again, as shown in Figure 3c. As a result, one selects nine as the optimal number of beams.

According to the results of the aforementioned analysis, the thickness of the front and rear beams has the greatest influence on the contact stress and fingertip displacement among the three geometric parameters. It is mostly because the finger surface is in direct contact with the fruit, and the thickness of the front and rear beams has a direct impact on the stiffness of the fingers. The structural parameters of the Fin Ray fingers are extremely complex, and this study just considers the most basic scenario. As a result, the best structural parameters are as follows: the thickness of the front and rear beams is 3.5 mm, the width of the fingers is 16 mm, and the number of beams is 9.

#### *2.2. Overall Design of the Soft Gripper*

The overall structure of the three-finger soft gripper for apple harvesting built with optimized Fin Ray fingers is shown in Figure 4a. It can be divided into three parts: the driving and sensing part, the transmission part, and the grasping part for clamping objects. The driving part is performed by a servo with torque and position feedback. To measure the relative distance between the gripper and the fruit, a distance sensor is mounted on the servo installation side of the gripper bottom plate. The transmission part is primarily accomplished by a slider, and the rocker mechanism was composed of a rocker, a connecting rod, a moving plate, and guide rods, as shown in Figure 4b. The servo rotates to drive the moving plate to move up and down. Because the fingers and their connectors are connected with the moving plate through the support rods, the fingers will move with the moving plate moving up and down, as shown in Figure 4c.

**Figure 4.** Overall design of the soft gripper: (**a**) overall structure; (**b**) details of transmission and the driving and sensing part; (**c**) the designed gripping mechanism; (**d**) details of the grasping part.

In the grasping part, three Fin-Ray finger units are evenly distributed around the bottom plate of the gripper disc, connected with the transmission mechanism by the finger connectors to drive the Fin Ray fingers. A silicone pad is attached to the surface of each finger to increase the contact friction between the finger and the fruit, which ensures the clamping stability, as shown in Figure 4d.

At the initial position, the finger connectors are inclined outward at a certain angle relative to the bottom plate. Because the bottom of the fingers is connected in parallel with the bottom of their connectors, and the finger has a triangular symmetrical structure, the clamping range of the gripper is expanded.

#### **3. Kinematic Mechanics Analysis of a Soft Gripper**

The driving force begins with the servo, travels through the slider and rocker mechanism, multi-link mechanism, and Fin Ray soft structure, and eventually acts on the gripped fruit. In conclusion, the static analysis of the rigid multi-link mechanism and the soft finger structure was performed to acquire the gripping force on the fruit surface. Simultaneously, the relationship between the gripper pulling force and the gripping force was investigated in connection with the pulling harvesting method. Because the three fingers are symmetrically arranged relative to the bottom plate of the gripper, and the structure is the same. Furthermore, the servo output torque operates on the center of the moving plate, and the movement process and stress situation are comparable. As a result, the stress analysis of

the direct contact between fingers and fruit begins with a single finger, making the analysis procedure simpler.
