**3. Mechanism Design of the Underactuated Manipulator**

#### *3.1. Determination of the Range of Flower Ball Surface Pressure and Stem-Cutting Force*

Broccoli picking involves two actions: enveloping and cutting. The enveloping action is performed by the clamping pressure of the underactuated mechanism, and stem separation is performed by the cutting tool. The maximum clamping pressure on the flower ball surface and the cutting force of the stem needs to be determined to provide a theoretical basis for the design of the rod length of the mechanical hand. The clamping action of the manipulator is composed of four contact forces: clamping pressure *F*<sup>1</sup> = *F*<sup>1</sup> and cutting force *F*<sup>3</sup> = *F*<sup>3</sup> (Figure 4).

**Figure 4.** Clamping workspace of the manipulator.

An LDW-1 universal material test machine and cutting tool (304 stainless steel; length, 80 mm; width, 50 mm; thickness, 2 mm; tool point angle, 8◦) was used. Thirty flowers with no surface damage were selected to determine the flower surface pressure and stem-cutting force, as shown in Figure 5.

**Figure 5.** Broccoli physical damage test: (**a**) Flower ball pressure test; (**b**) Stem-cutting force test.

The dynamic compression mold of the universal material test machine was operated at a uniform speed of 10 mm/min. Different loads (30, 40, and 50 N) were applied to the surface of the flower ball, and the damage degree was compared and analyzed, as shown in Figure 6.

When the pressure was 30 N, the surface of the flower ball had no obvious damage. When the pressure was 40 N, small flower buds on the compressed surface of the flower ball were slightly deformed, and the small flower buds on the compressed part were damaged. When the pressure was 50 N, the small stem on the inner surface of the flower ball under pressure was deformed, and the small bud under pressure was damaged seriously. Therefore, the clamping pressure was maintained at 25–30 N, which can ensure the non-destructive enveloping of the flower ball.

The fixture of the moving platform on the test machine drove the cutting tool to move at uniform speeds of 30, 80, and 100 mm/min, and 30 groups of cutting tests were carried out. When the cutting blade was in contact with the broccoli stem, the force sensor on the test machine could detect the cutting force in real-time. The cutting force statistics are shown in Figure 7. Each dot in Figure 7 represents the maximum stem-cutting force for each set of tests. In the 30 groups of stem-cutting force, the cutting force represented by red dots was less than that of blue dots and more than that of green dots. The blue dot is the maximum cutting force of 37.28 N, the green dot is the minimum cutting force of 28.36 N, and the average cutting force is 34.89 N. According to the above cutting force

(**a**) (**b**) (c)

characteristics, keeping the cutting force at 30 N–35 N can ensure that the stems can be cut and separated.

**Figure 6.** Comparison of pressure damages on broccoli surfaces at different pressures: (**a**) 30 N; (**b**) 40 N; (**c**) 50 N.

**Figure 7.** Stem-cutting force test statistics.

#### *3.2. Kinematics Analysis of the Underactuated Mechanism*

As shown in Figure 8, the underdrive mechanism designed in this paper is bilaterally symmetric. Therefore, one side of the mechanism was analyzed. When the underactuated mechanism was not in contact with the target object, the relative position of each linkage in the mechanism remained unchanged under the constraint of the torsional spring. The meaning of specific mathematical symbols can be found in Table 1. The specific meanings of mathematical symbols in Figure 8 can be referred to in Table 1. The red characters *θ*1, *θ*2, and *θ*<sup>3</sup> in Figure 8 respectively represent angle variations of rods *AB*, *AD*, and *DK*.

**Figure 8.** Motion analysis of the underactuated mechanism.



For the velocity at the contact points of each joint, the following formula can be obtained according to the projection theorem of the velocity of the rigid body plane motion:

$$\begin{cases} \begin{array}{c} \boldsymbol{v} = \begin{bmatrix} \boldsymbol{v}\_{1} \\ \boldsymbol{v}\_{2} \end{bmatrix} = \begin{bmatrix} & \boldsymbol{l}\_{1} & \boldsymbol{0} \\ \boldsymbol{l}\_{2} + d \cos \theta\_{3} & \boldsymbol{l}\_{2} \end{bmatrix} \begin{bmatrix} \dot{\theta}\_{2} \\ \dot{\theta}\_{3} \end{bmatrix} \\\ J\_{\mathbf{v}} = \begin{bmatrix} & \boldsymbol{l}\_{1} & \boldsymbol{0} \\ \boldsymbol{l}\_{2} + d \cos \theta\_{3} & \boldsymbol{l}\_{2} \end{bmatrix} \end{cases} \tag{1}$$

The four-bar mechanism has the following form:

$$\begin{cases} \begin{bmatrix} \dot{\theta}\_2 \\ \dot{\theta}\_3 \end{bmatrix} = \begin{bmatrix} 1 & -A \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \dot{\theta}\_1 \\ \dot{\theta}\_3 \end{bmatrix} \\\ J\_{uv} = \begin{bmatrix} 1 & -A \\ 0 & 1 \end{bmatrix} \end{cases} \tag{2}$$

where *A* is shown in the following equations:

$$\begin{cases} \begin{aligned} A &= \frac{(\sqrt{1-B} + \sin \varphi\_2)\sqrt{1-C}}{\sqrt{1-B}(\sqrt{1-C} + \sin \varphi\_1)} = 1 \\ B &= \frac{\left[a^2 + b^2 - c^2 + 2cd\cos \varphi\_2 - d^2\right]^2}{4a^2b^2} = \cos^2 \varphi\_2 \\ C &= \frac{\left[c^2 + b^2 - a^2 + 2ad\cos \varphi\_1 - d^2\right]^2}{4b^2c^2} = \cos^2 \varphi\_1 \end{aligned} \end{cases} \tag{3}$$

Therefore, when the slider *E* moves down to drive the drive rod *BG* to move, the underdrive mechanism rotates around base point *A*. When the clamping guard plates contact the surface of the flower ball, the rocker *AD* stops moving. The driving force drives the swing rod *DK* to move and closes the cutting blade to overcome the binding force of the torsional spring and complete the clamping action.
