*3.1. Force Analysis of Rigid Multi-Link*

The basic structure and motion principle of the soft gripper is shown in Figure 5a,b. The force acting on the fruits of the Fin Ray structure can be equivalent to a single concentrated force in the analysis of the rigid multi-link (the analysis of the soft Fin Ray structure will be discussed below). The servo drives the rocker to rotate counterclockwise when grabbing, the moving plate to travel down along the guide rod, and the support rod to move. Following that, the support rod drives the finger connector to rotate around the joint FF, resulting in the envelope-gripping movement of the finger.

**Figure 5.** Motion schematic of the gripper: (**a**) physical model; (**b**) kinematics model.

In the figure: *Md* is the servo output torque; *θFR* is the angle between the rocker and the horizontal direction; *θFF* is the angle between the finger connector and the horizontal direction; *α* is the angle between the front and rear beams of fingers and the base.

Because of the light weight of each moving part of the rigid multi-link, the gravity and inertia force during the movement of the gripper can be ignored.

The mechanical analysis of the multi-link mechanism is performed under static equilibrium conditions. The connecting rod is vertical to the moving plate at the time of initial contact. Their angle does not alter much when the rocker is rotated. To make the calculation easier, the difference is negligible. Among the multi-link, the connecting rod is a two-force member, and the moving plate is employed to assess the force, as shown in Figure 6. Therefore, one has

$$F\_{\mathbb{C}M} = F\_{\text{MS}\prime} \tag{3}$$

**Figure 6.** Force analysis of the moving plate.

For which, *FCM* and *FMS* are in the opposite direction. *FXY* is the force of member X applying to member Y. To simplify the analysis, the sliding friction between the moving plate and the guide rod is negligible.

Thus, the support rod is a two-force member. Figure 7a shows the force analysis of the finger and its connector. The closing force triangle shown in Figure 7b can be obtained according to the geometric conditions for the equilibrium of the plane-intersecting force systems.

**Figure 7.** Force analysis of the finger and its connector: (**a**) force diagram; (**b**) closing force triangle.

To maintain the force balance of the finger and its connector, one obtains

$$F\_{\mathcal{E}}\cos\gamma = F\_{\mathcal{F}\mathcal{F}}\cos\beta\_{\prime} \tag{4}$$

$$F\_{\mathcal{C}}\sin\gamma + F\_{FF}\sin\beta = F\_{SF} \tag{5}$$

where *Fc* is the contact reaction between the finger and fruit, that is the finger gripping force; *γ* is the angle between *Fc* and the horizontal direction; *β* is the angle between *FSF* and the horizontal direction.

According to Equations (4) and (5),

$$F\_{\mathbb{C}} = \frac{1}{\sin \gamma + \cos \gamma \tan \beta} F\_{\, SF \, \prime} \tag{6}$$

where

$$
\gamma = \frac{\pi}{2} - \mathfrak{a} + \theta\_{FF} \tag{7}
$$

$$\tan\beta = \frac{h}{L\_{Fc}\cos\theta\_{FF}\sin^2(\alpha - \theta\_{FF})} - \frac{1}{\tan(\alpha - \theta\_{FF})},\tag{8}$$

where *h* is the distance from the center of fruit to the bottom plate of the gripper; *LX* is the length of component X, that is *LFc* is the length of the Finger connector, and *LR* is the length of the rocker.

To obtain the relationship between the servo torque *Md* and the gripping force *Fc*, the rocker is taken as the forced object, and the force situation is shown in Figure 8. The moment balance at joint FR is

$$F\_{\rm CR} L\_R \cos \theta\_{FR} = M\_d. \tag{9}$$

**Figure 8.** Force analysis of the rocker.

According to the force characteristics of the two-force members,

$$F\_{\rm CR} = F\_{\rm MS} = F\_{\rm FS} = F\_{\rm SF} \tag{10}$$

$$\text{where } F\_{FS} \text{ and } F\_{SF} \text{ are in opposite directions.}$$

From Equations (6), (9), and (10), one can obtain

$$F\_{\rm c} = \frac{1}{\sin \gamma + \cos \gamma \tan \beta} \cdot \frac{1}{L\_R \cos \theta\_{FR}} \cdot M\_d. \tag{11}$$

#### *3.2. Contact Force Analysis between Soft Finger and Fruit*

When the finger comes into contact with the fruit, it creates an adaptable envelope, and the contact area expands. The flexible deformation of the Fin Ray structure makes the mechanical analysis difficult. Therefore, to facilitate the calculation, the fruit is simplified as a regular sphere. Aiming at the picking method for pulling fruits, a simplified single-finger plane stress model is given in Figure 9.

**Figure 9.** Plane force model of fruit (the forces are shown in red when the load divisions on the x-axis are in the same direction as the x-axis and black in the opposite direction).

The contact between the finger and the fruit is divided into two areas with angles of *δ* and *σ*, with the y-axis as the limit. The positive touching pressure of the fruit is simplified as a uniform load; the size is *n*; the unit is N/m, and the directions all point to the center of the circle, whose angle with the y-axis is *ϕ<sup>i</sup>* (*i* = 1, 2, ... , m). *Fs* is the static friction force generated by the positive pressure of the finger on the fruit. When pulling the fruit, the positive pressure on the fruit and the component force of the static friction force generated along the x-axis direction are the main forces to ensure the stability of grasping. Specify that the direction of the force is positive along the positive x-axis. To obtain the resultant force in the x-axis direction *F*, one has

$$\begin{array}{lcl}F&=\int\_{-\delta}^{\sigma} (F\_s \cos \varrho + n \sin \varrho) \cdot r d\varrho\\&=r \cdot [F\_\delta (\sin \delta + \sin \sigma) + n(\cos \delta - \cos \sigma)]\_\prime \end{array} \tag{12}$$

where

$$F\_s = \mu \cdot n.\tag{13}$$

In the Equation, *μ* is the maximum static friction coefficient between the finger and the pericarp of the fruit; *r* is the radius of the fruit.

Therefore, from Equations (11) and (12), the relationship between the resultant force *F* and the positive touching pressure on the fruit can be obtained,

$$F = rn \cdot \left[ \mu (\sin \sigma + \sin \delta) + (\cos \delta - \cos \sigma) \right]. \tag{14}$$

The relationship between the equivalent single concentrated force *Fc* and the uniform load *n* in the rigid multi-link force analysis above is

$$F\_{\mathfrak{c}} = \int\_{-\sigma}^{\delta} n \cdot r d\psi = nr \cdot (\delta + \sigma). \tag{15}$$

According to Equations (11), (14), and (15), the relationship between the servo torque *Md* and the resultant force *F* can be obtained as

$$F = \frac{\mu \cdot (\sin \sigma + \sin \delta) + (\cos \delta - \cos \sigma)}{L\_R \cdot (\sin \gamma + \cos \gamma \tan \beta) \cdot \cos \theta\_{FR} \cdot (\delta + \sigma)} \cdot M\_d. \tag{16}$$
