3.1. Static analysis of adaptive clamping belts

A second-order Yeoh model is based on the following assumptions: 1) the silicone rubber material is homogeneous and incompressible and 2) the flattened metal clamping layer is not extensible.

The second-order strain energy density function for the Yeoh model is defined by

(1) \begin{equation}W=C_{10}\left(I_{1}-3\right)+C_{20}({I_{1}}-3)^{2}\end{equation}

where W is the strain energy, I ₁ is the first invariant of the deformation tensor, and C ₁₀ and C ₂₀ are the parameters of the second-order Yeoh model.

According to the isotropic assumption of rubber theory, the strain energy density function can be used to describe its mechanical properties. The relationship between the strain energy density function W and the three invariants of the deformation tensors I ₁, I ₂, and I ₃ is

(2) \begin{equation}W=W(I_{1},I_{2},I_{3})\end{equation}

The relationship between the three invariants of the deformation tensors I ₁, I ₂, and I ₃ and the elongation ratios λ ₁, λ ₂, and λ ₃ in the three directions of the airbag is

(3) \begin{equation}I_{1}={\lambda _{1}}^{2}+{\lambda _{2}}^{2}+{\lambda _{3}}^{2}\end{equation}

(4) \begin{equation}I_{2}={\lambda _{1}}^{2}\cdot {\lambda _{2}}^{2}+{\lambda _{1}}^{2}\cdot {\lambda _{3}}^{2}+{\lambda _{2}}^{2}\cdot {\lambda _{3}}^{2}\end{equation}

(5) \begin{equation}I_{3}={\lambda _{1}}^{2}\cdot {\lambda _{2}}^{2}\cdot {\lambda _{3}}^{2}=1\end{equation}

Table I. Mechanical properties of the silicone rubber materials-60 degree.

In the uniaxial tensile test, assuming that the one direction is the main tensile direction, $\lambda _{2}=\lambda _{3}$ and $\lambda _{1}=\lambda$ . Substitution formula (5):

(6) \begin{equation}\lambda _{1}={\lambda _{2}}^{-2}={\lambda _{3}}^{-2}=\lambda\end{equation}

A total of eight sets of uniaxial tensile experiments were performed on the silicone rubber materials-60 degree, and the resulting experimental data were fitted to the data fitting function based on nonlinear least squares fitting. Each sample was tested eight times, and the average value was taken, so C ₁₀ = 0.05 MPa and C ₂₀ = 0.01 MPa were obtained.

The initial state of the adaptive clamping belt can be simplified into a set of linearly arranged independent meshes (Fig. 4(a)). In order to control the bending rate during airbag inflation, it is necessary to build a set of bending models with independent meshes to obtain the equation of bending angle versus input air pressure.

Figure 4. Schematic diagram of the deformation of a set of independent meshes. (a) Initial status, (b) after inflation.

During the inflation process, according to the structure of the adaptive clamping belt, the drive layer will deform unidirectionally under the constraint of the limiting layer, and the final outer wall of the front end of the drive layer can be viewed as a circular arc with radius r (Fig. 4(b)). The deformation of a single grid angle is $\alpha$ . The whole group has a total of i independent grids, and assuming that the deformation of each independent grid angle is equal, the deformation of the whole group of grid angles is $=i\cdot \alpha$ . The angle of the center of the circle after bending the driving layer is $\beta$ . By simulating the deformation of a set of independent meshes and averaging it over several simulations in the ANSYS software, $\beta =4\alpha$ is obtained.

The midpoint of the front outer wall of the independent grid of the drive layer is used as the coordinate origin to establish a right-angled coordinate system (Fig. 4(b)), and the coordinates of the circular arc of the front outer wall of the inflated drive layer can be expressed as

(7) \begin{equation}x^{2}+(y-{y_{0}})^{2}=r^{2}\!\left(-\frac{d}{2}\leq x\leq \frac{d}{2},y\lt 0\right)\end{equation}

where the variables x and y represent the horizontal and vertical coordinates, respectively, of the point on the arc of the outer wall of the front end of the drive layer after inflation of the drive layer, $r=d/[2\sin (\beta /2)]$ and $y_{0}=d/[2\tan (\beta /2)]$ .

In the initial state, the thickness of the driving layer is H, and the thickness of the connecting layer is h. During the inflation process, the drive layer expands, resulting in the connecting layer being gradually stretched and thinned. According to the Boltzmann distribution law and the structure of the adaptive clamping belt, the force distribution of the driving layer and the connecting layer is uniform and equal in size, and the main elongation λ ₁ is related to the centroid angle $\beta$ after bending of the driving layer by the equation (8):

(8) \begin{equation}\lambda _{1}=\frac{\beta }{2\sin \left(\frac{\beta }{2}\right)-2\beta \cdot \sin (\frac{h}{2})}\end{equation}

The initial length of the driving layer is L, and the length d after deformation can be expressed as

(9) \begin{equation}d=2\!\left[2\lambda _{1}\cdot \left(2h+H\right)\cdot \sin \left(\frac{h}{2}\right)+4\right]\end{equation}

Assuming that the deformation of the outer wall of the drive layer also increases linearly to y(x) during the increase in air pressure from 0 to p, the work acting on the outer wall can be expressed as

(10) \begin{equation}U\left(p,\beta \right)=\int _{0}^{d}p\cdot y\left(x\right)dx\end{equation}

Therefore, the strain energy density function can be expressed as

(11) \begin{equation}W\left(\beta \right)=C_{10}\!\left(\lambda ^{2}+\frac{1}{\lambda ^{2}}-2\right)+C_{20}\!\left(\lambda ^{2}+\frac{1}{\lambda ^{2}}-2\right)\end{equation}

The deformation energy after deformation of the driving layer can be expressed as

(12) \begin{equation}V\left(\beta \right)=H\cdot L\cdot W(\beta )\end{equation}

According to the law of conservation of energy, the work acting on the outer wall of the drive layer is equal to the deformation energy of the drive layer after deformation; therefore:

(13) \begin{equation}U\left(p,\beta \right)=V(\beta )\end{equation}

The relationship between p and $\alpha$ can be introduced through the relationship between p and $\beta$ . By associating equations (7)–(13), the relationship equation between the whole set of mesh bending angles θ and the input air pressure p can be obtained as follows:

(14) \begin{equation}\theta =i\cdot \alpha =i\cdot f(p)\end{equation}

According to the established mathematical model, an independent mesh bending trend is obtained (Fig. 5). When the input air pressure reaches 70 kPa, the bending angle is about 56°, and the subsequent growth of the bending angle is gradually flattened with the increase in air pressure.

Figure 5. Variations of the drive bending angle with the pneumatic pressure.

3.2. Finite element simulation analysis of the drive layer

In order to verify the feasibility of multidirectional and multiangle bending of the adaptive clamping belt, simulation analysis was conducted on the driving layer using ANSYS finite element software. The Yeoh model of second-order hyperelastic material is selected; set C ₁₀ = 0.05 MPa and C ₂₀ = 0.01 MPa; tetrahedral mesh division is used; boundary conditions are set as fixed constraints; and air pressure perpendicular to the surface of the internal chambers of the drive layer is applied without taking into account the role of compressed air at the exit of the drive layer. Without considering gravity and friction, the bottom surface of the airbag is fixed to the flat metal clamping layer by means of a flexible protective sleeve. As shown in Fig. 6, the deformation of the independent grids of the drive layers in the adaptive clamping belt under different input air pressures is shown, divided into 8 levels, and the input air pressure difference of 10 kPa in each stage is shown. As the input air pressure increases, so does the bending angle of the independent mesh of the drive layer. In the process of increasing the input air pressure, the bending angle change rate of the independent grid is getting faster and faster, but when the input air pressure exceeds 40 kPa, the bending angle change rate of the independent grid gradually slows down. When the input air pressure reaches 70 kPa, the independent grid bending angle of the driving layer is 52° under the constraint of the limit layer. The input air pressure continues to increase, but the bending angle of the independent mesh hardly changes. From the input air pressure of different levels, it can be seen that the inner side of the independent mesh is more stressed than the outer side, and the outer side of the independent mesh is more deformed than the inner side (the inner side of the independent mesh is the side close to the limit layer, and the outer side of the independent mesh is the side close to the connecting layer), which verifies the accuracy of the mathematical model built in Section 3.1.

Figure 6. Simulation results of the driver layer.

3.2. Dynamic simulation analysis of gripper for casting sorting robot

In order to more clearly reflect the dynamic characteristics of the rigid–flexible coupling model of the casting sorting robot gripper. It is necessary to carry out mesh delineation of the adaptive clamping belt by the ANSYS software and then import the corresponding rigid components into the ADAMS software to replace them, so as to establish the rigid–flexible coupling model of the casting sorting robot gripper (Fig. 7).

Figure 7. Rigid–flexible coupling dynamics model of the gripper of casting sorting robot.

Due to the complexity of the gripper structure of the casting sorting robot, in order to more accurately reflect the movement status of the gripper during the working process, it is necessary to simplify and process the gripper model appropriately. First of all, the parts that do not participate in the transmission inside the gripper are boolean operations. Second, the dimensional tolerances and assembly errors of the model are not considered. Third, in the rigid–flexible coupling model of the gripper, except for the adaptive clamping band that was flexibly processed by ANSYS, the rest of the components were treated as rigid bodies.

In order to test the maximum diameter of the gripper to cope with the castings with complex shapes, the current motion simulation of the gripper is now divided into two main working conditions for research. First of all, the materials of each component of the gripper are defined, in which the active and driven support chains are made of aluminum alloy, and the support chain is made of steel. The mechanical properties are illustrated in Table II. Secondl, the casting sorting robot gripper virtual prototype components add a fixed pair between the support of the motion pair added to the clamping virtual prototype component and the earth, the support and the active support chain in the clamping cylinder, the support and the follower chain of the upper linkage to add the rotation of the vice, the active support chain in the clamping cylinder and the piston rod to add the moving vice between the piston rod and the drive, and this moving vice to add. The driving speed of the piston rod is set to 6 mm/s, the simulation time is 8 s, and the simulation step is 0.001 s.

Simulation condition 1: The gripper carries out adaptive wrapping and gripping operations on a rigid casting with a rectangular body at one end and a hexagonal steel at the other end. The state of the simulation process is shown in Fig. 8.

Figure 8. Rigid–flexible coupling dynamics model of the gripper of casting sorting robot under working condition 1.

Simulation condition 2: The gripper carries out adaptive wrapping and gripping operations on rigid castings that are cylinders at one end and circular table bodies at the other end. The state of the simulation process is shown in Fig. 9.

Figure 9. Rigid–flexible coupling dynamics model of the gripper of the casting sorting robot under working condition 2.

According to the ADAMS simulation results, the maximum diameter of the casting sorting robot gripper for stable gripping is 140 mm.

5.1. Bending performance experiments of adaptive clamping belt

Some of the states of the casting sorting robot gripper during the gripping motion are shown in Fig. 13. Combined with the designed pneumatic control system, different external pneumatic loads can be applied to each clamping cylinder, resulting in multiangle bending of the adaptive clamping belt as a whole and realizing the diversity of wrapping postures. Each adaptive clamping belt can be controlled independently during the experiment. Gripping object diversity is achieved by applying different external air pressure loads to each airbag in the drive layer of the adaptive clamping belt.

Figure 13. Adaptive clamping belt bending performance test. (a) Single adaptive clamping belt bending test, (b) adaptive clamping belt bending experiment under no-load condition.

5.2. Comparative performance experiments of casting sorting robot grippers

The designed casting sorting robot gripper, in the process of gripping the workpiece, first drives the adaptive clamping belt through the active support chain to complete the preliminary wrapping of the gripped workpiece and then inflates the airbag of the driving layer to make the adaptive clamping belt tightly squeeze and wrap the workpiece under the constraints of the limiting layer. For the existing rigid mechanical gripper, due to its lack of adaptability, it usually cannot smoothly grasp shaped workpieces, and it is easy to damage the surface of the workpieces during the gripping process. The soft gripper is usually used for gripping lightweight objects, and its load capacity is low.

As shown in Fig. 14, the designed casting sorting robot gripper and rigid mechanical gripper are used to carry out grasping test comparison experiments. During the experimental process, the designed casting sorting robot gripper successfully grasps the caliper bracket casting in 1s (Fig. 14(a)), while the rigid mechanical gripper slips after grasping for 0.4 s and is unable to smoothly grasp the caliper bracket casting (Fig. 14(b)). For the soft mechanical gripper, although it can complete the wrapping of the caliper bracket castings, it cannot bear the weight of the caliper bracket castings, which makes the gripper excessively deformed.

Figure 14. Comparative experiments of casting grasping. (a) The designed casting sorting robot gripper gripping the caliper bracket casting, (b) rigid manipulator gripping the caliper bracket casting.

By analyzing the common small- and medium-sized castings, we selected six types of workpieces with a mass range of 100–1000 g to replace the castings for the gripping experiment. As shown in Fig. 15, the selected workpieces are closer to the small- and medium-sized castings in terms of their hardness, surface roughness, and other physical properties. By using the designed casting sorting robot gripper, rigid mechanical gripper, and soft body manipulator for each of the six types of workpieces, 10 gripping tests were conducted, and the results are shown in Fig. 16. The designed casting sorting robot gripper can successfully complete the gripping test of six types of workpieces, and the comprehensive success rate is as high as 96.4%. The gripping success rate is much higher than that of the same specifications of the rigid mechanical gripper and the soft robotic hand. To a certain extent, it solves the existing small- and medium-sized castings sorting robot gripper self-adaptive insufficient problem, but also for all kinds of shaped workpiece clamping robot design and optimization provides a reference.

Figure 15. Gripped workpieces. (a) Cylinder, (b) cuboid, (c) sphere, (d) bracket, (e) plier, (f) cone, (g) pipe joint, (h) shovel, (i) hexagonal steel, (j) pipe wrench.

Figure 16. Number of successful grabs for each gripper.

5.3. Adaptation experiment of casting sorting robot gripper

In order to verify the adaptability and load capacity of the gripper of the casting sorting robot, we carried out adaptive experiments by gripping the workpieces shown in Fig. 15 in turn, and the parameters of the workpieces are shown in Table III. In the process of gripping the workpiece, first increase the pressure of the clamping cylinder in the active support chain, and after the adaptive clamping belt has completed the initial wrapping of the workpiece, inflate the airbag of the adaptive clamping belt to tightly squeeze and wrap the gripped workpiece. As shown in Fig. 17, if the workpiece does not fall during the gripping process, the gripping is considered successful.

Table III. Parameters of the gripped workpieces.

²Note: The wrapping angle refers to the central angle between the adaptive clamping belt and the contact arc of the workpieces when the casting sorting robot gripper completes the adaptive wrapping and stabilizes the clamping of the workpiece.

Figure 17. Gripper for gripping the workpieces. (a) Gripping a cylinder, (b) gripping a cuboid, (c) gripping a sphere, (d) gripping a bracket, (e) gripping a plier, (f) gripping a cone, (g) gripping a pipe joint, (h) gripping a shovel, (i) gripping a hexagonal steel, (j) gripping a pipe wrench.

The gripper has a good fit and stable gripping of all kinds of small- and medium-sized workpieces, as shown in Fig. 15, which proves that the shape of the gripped workpieces will not limit the gripping ability of the gripper. The maximum load capacity of the gripper was experimentally measured to be 930 g, and the maximum wrap angle was 296°, and it still has some upward space.

5.4. Characteristics of the designed gripper for casting sorting robots

In the actual production process of small- and medium-sized castings, the shape of the casting is often irregular, with more complex structures. Existing grippers are limited by the low adaptability of rigid grippers and the low load capacity of soft grippers when performing gripping tasks, which makes it difficult to meet various demands at the same time. In order to solve the problem of the insufficient adaptability of existing small- and medium-sized casting sorting robot grippers, we designed a casting sorting robot gripper with a rigid–flexible coupling structure. The mechanical design of the gripper imitates the gripping posture of the human hand; with its compact structure and flexible gripping, it can be applied to wrapping and gripping all kinds of small- and medium-sized castings. The gripper adopts the pneumatic dual-control system method to ensure that the casting is firmly clamped without damaging the casting, which improves the working efficiency of the gripper. Compared with existing rigid manipulators, soft manipulators, and variable stiffness manipulators, the casting sorting robot gripper we designed can realize high-efficiency and high-performance adaptive gripping with the stability of a rigid manipulator and the suppleness of a soft manipulator, while ensuring that the weight of the casting is less than 930 g and the diameter is less than 140 mm. In order to ensure gripping reliability, the designed gripper is not suitable for gripping castings with a thickness of less than 15 mm or a length of less than 100 mm, as this would result in the adaptive clamping belt not being able to fit snugly around the gripped casting.

In order to more intuitively represent the characteristics of the casting sorting robot gripper, we compared the designed casting sorting robot gripper with existing similar designs, as shown in Table IV.

Table IV. Comparative analysis of the performance of similar grippers.