*3.1. Establishment of the Single-Finger Kinematics Model and Setting of Key Component Parameters*

According to the size statistics of cherry tomatoes planted in the standardized greenhouse, their diameters are about 20~40 mm. The kinematics model of the gripper was established first to meet the size requirements, then the parameters of relative parts were determined. Due to the gripper consists of three symmetrical fingers, the single finger was modeled and analyzed. Figure 4 shows the schematic representation of the structural parameters of gripper's the single finger. The rectangular coordinate systems *xoy* and *XOY* were created. The center of the chassis was set to the origin of *XOY*. The connection between the fingers and the chassis were set to the origin of *XOY*. The axes were established in the direction of pole motion and perpendicular to it, respectively.

**Figure 4.** Schematic diagram of the motion structure parameters of gripper's the single finger. Note: Curve *OB* is the initial position of the finger; *A* and *A*' represent the contact point between the constraint part and the finger before and after polymerization, respectively; *B* and *B*' represent the position of the finger end before and after polymerization, respectively; Arc *BB*' is the motion trajectory of the finger end; *θ* is the rotation angle of the finger; *h0* is the distance from the finger bottom to the center of the chassis; *R* is the distance from the finger bottom to end.

According to the grasping characteristics, the opening and closing of the fingers could be regarded as the rotation around *O*. *BB*' was the circular arc with *O* as the center and *R* as the radius. *Q* and *Q*' were set to random points on the *OB* and *OB*', respectively. The homogeneous coordinates of *Q* in the *xoy* coordinate system is:

$$Q\_{xoy} = \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = T \times R \times Q\_{XOY} \tag{2}$$

where *QXOY* = ⎡ ⎣ *X Y* 1 ⎤ ⎦ is the homogeneous coordinate of *Q* in *XOY*; *R* = ⎡ ⎣ cos *θ* sin *θ* 0 − sin *θ* cos *θ* 0 0 01 ⎤ ⎦ ⎡ 10 0 ⎤

is the rotation matrix; *T* = ⎣ 0 1 *h*<sup>0</sup> 00 1 ⎦ is the translation matrix of the position of the *XOY*

relative to *XOY*. After rotating *θ* clockwise, the equation for *OB*' under *XOY* is:

$$X\sin\theta + Y\cos\theta = 12\sin\left[\frac{1}{30}(X\cos\theta - Y\sin\theta - 20\pi)\right] + 6\sqrt{3}, \ \theta \in [0, 45\pi\cos\theta] \tag{3}$$

The coordinate of the *Q*' on the *OB*' in *XOY* is:

$$Q'\_{\mathbf{x}\mathbf{y}} = \begin{bmatrix} \mathbb{x}\_{Q'} \\ \mathbb{y}\_{Q'} \end{bmatrix} = \begin{bmatrix} X\cos\theta + Y\sin\theta \\ h - X\sin\theta + Y\cos\theta \end{bmatrix} \tag{4}$$

When the movement of the constraint part caused the finger to rotate *θ*, *XA* was always on the *x*-axis and *y* = 0. The movement distance of constraint part *XA* was set to *L*. From Equation (3), the relationship between *θ* and *L* can be expressed as a complex nonlinear equation, which in MATLAB can be described by the solve function as:

$$f(\theta, m) = L \sin \theta - 12 \sin \left[ \frac{1}{30} (L \cos \theta - 20 \pi) \right] - 6 \sqrt{3}, \ \theta \in [0, 45 \pi \cos \theta] \tag{5}$$

The *n*, the open range of the finger end, can be expressed as:

$$\begin{cases} \begin{array}{c} n = 2y\_{B'} \\ y\_{B'} = h - X\_B \sin \theta + Y\_B \cos \theta \end{array} \tag{6} \end{cases} \tag{6}$$

Figure 5 was inferred from Equations (5) and (6). With the increase in *L*, *θ* and *n* changed synchronously, and showed a negative correlation. As the constrain part advanced, the finger's movement could be divided into three stages. Stage one was the constraint part did not touch the finger, and *θ* and *n* did not change. Stage two was a gripper closing stage. As the constraint part advanced to the *LC*, it touched the finger and caused it to start closing. Then, *L* increased, *θ* increased, and *n* decreased. When the constraint reaches *LD*, *θ* and *n* reached the maximum and minimum, respectively. Stage three was gripper opening stage. As the constraint part pass through *LD*, *θ* decreased and n increased.

**Figure 5.** The relationship between *L* and *θ* and *n.* Note: C and D are the two points through which the constraint part moves from chassis to finger ends. -<sup>1</sup> , -<sup>2</sup> and -3 are three stages divided by the changes in *n* and *θ*.

Therefore, stage one can be used as the stopping region of constraint the part in the natural state. The constraint part does not encounter fingers, so that the parts can be well protected. In stage two, the gripper end had the maximum and minimum gripping range. This stage can be used as the movement region of the constraint part in the grasping process. Stage three cannot be used as the region of movement of the constraint part.

Therefore, stage two was discussed to adapt the cherry tomato sizes. From Equation (1), when *y* = 0, *LC* = 31.4 mm, *θ*min= 0◦, and *nmax*= 2*h*+20.78 mm. Equation (5) was calculated by the chain rule as:

$$z = \frac{\mathbf{d}\theta}{\mathbf{d}L} = -\frac{\mathbf{F}\_L'}{\mathbf{F}\_\theta'} \tag{7}$$

Let *z* = 0, and *L* can be calculated as:

$$L = \frac{30}{\cos \theta} \left( \frac{2\pi}{3} + \arccos \frac{5 \sin \theta}{2 \cos \theta} \right) \tag{8}$$

When the above formula was associated with *f*(*θ*, *m*)= 0, *LD* = 94.64 mm, *θmax* = 12.45◦, and *n*min= 2*h*−40.64 mm. Thus, the range of *n* was 2*h*−40.64 mm to 2*h*+20.78 mm. The open range of the finger end was only related to *h*. Therefore, when *h* was determined to be 20.4 mm, the open range of the gripper end was about 0~61.6 mm. It can meet the sizes requirement of cherry tomato picking.

According to the above analysis, the length of a single finger was 140 mm. As the fingers were printed in 3D, the thicknesses were set to 5 mm, and the widths were set to 10mm for grasping reliability. The length of the pole should be greater than 94.64 mm because the active regions of the constraint part were stage one and two. Therefore, the length of the pole was set to 120 mm because it was to be fixed to the slider. The length of screw was set to 150 mm for security during system operation. The pitch was 10 mm. The interior radius of the constraint part was set to 21 mm to ensure that the finger reaches the above grabbing range, which is slightly larger than *h*. Outer radius was set to 25 mm. The Radius of the Chassis was set to 50 mm to mesh with the gears of the DC motor. The width of steel shrapnel is the same as the width of the finger. The parameters of key parts of the end-effector were determined according to *h*. (Table 1).


**Table 1.** The table of the key parts parameters.
