# Effects of Flexural Rigidity on Soft Actuators via Adhering to Large Cylinders

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Design and FEMs of the SPAA

#### 2.1. Design of the SPAA

#### 2.2. FEMs of the Bending Curvature and Flexural Rigidity

_{10}= 0.4138, C

_{01}= 0.1034, d = 2) and the Arruda–Boyce model [25] (μ = 0.5788, λ

_{m}= 1.2099) were used to characterize the top elastic actuator and the bottom foam rubber, respectively.

_{1}~C

_{4}to characterize the bending curvature of PN1~PN4, respectively. We divided the abdomen of the SPAA into four equal parts with five yellow markers (numbered marker0~marker4 from the root to the end), recorded the simulation results of the five markers’ coordinates under the SPAA’s bending state, and obtained C

_{1}~C

_{4}according to C = α/l Where α and l, respectively, represent the central angle and length of the arc that each PN bends into under positive pressure. Specifically, α can be obtained by geometric calculation based on the recorded data of markers’ coordinates (the angles between the connecting line of Marker0 and Marker1~Marker4 and the vertical direction were numbered as θ

_{1}~θ

_{4}, respectively, then α

_{1}~α

_{4}were 2θ

_{1}, 2(θ

_{2}− θ

_{1}), 2(θ

_{3}− θ

_{2}), 2(θ

_{4}− θ

_{3}), respectively). And l = 12 mm. The simulation results of four PNs’ bending curvatures under the pressures varying from 0 to 100 kPa at 10 kPa intervals (Figure 2b) show that C

_{1}~C

_{4}all increase with an increase in the air pressure, and the relationships are approximately linear (${R}_{{C}_{1}}^{2}=0.9974,\text{}{R}_{{C}_{2}}^{2}=0.9978,\text{}{R}_{{C}_{3}}^{2}=0.9963,\text{}{R}_{{C}_{4}}^{2}=0.9970$). However, C

_{1}is obviously lower than C

_{2}~C

_{4}. It is because one side of PN1 is directly fixed to the root, thus limiting the bending of the PN1. As for PN2~PN4, the bending curvature is similar. Therefore, we used the mean value of the C

_{2}~C

_{4}to define the bending curvature C of the SPAA. The simulation results of bending curvatures of three SPAAs with different rib heights under pressures varying from 0 to 100 kPa at 10 kPa intervals (Figure 2c) show that bending curvatures of SPAAs are positively correlated with the pressure, and the relationships are approximately linear (${R}_{h=4}^{2}=0.9957,{R}_{h=6}^{2}=0.9980,\text{}{R}_{h=8}^{2}=0.9996$). The results indicate that the SPAA can continuously bend with controllable bending curvature under positive air pressures.

^{2}, an increase of 120%, which is much higher than the 10% increase in the flexural rigidity generated by decreasing the width of the rib from 3 to 0.8 mm [21]. The results indicate that it is theoretically reasonable to significantly increase the flexural rigidity of the SPAA by increasing the height of the rib.

## 3. Experimental Setup for Characterizing the Performance of the SPAA

#### 3.1. Fabrication of the SPAA

#### 3.2. Setup of the Synchronous Testing Platform

## 4. Results

#### 4.1. Bending Curvature and Flexural Rigidity

^{−1}, respectively. It can be verified that the SPAA can continuously bend under positive air pressure drive and the bending curvature is controllable. Compared to the rigid [9,28] or under-actuated [10] soft actuators, SPAAs can actively envelop a wide range of cylinders with different curvatures. The simulation results in Figure 2c deviate within 7% of the experimental results.

^{2}, an increase of 117%, demonstrating that it is more reasonable to significantly increase the flexural rigidity of the SPAA by increasing the rib height than decreasing the rib width [21]. The above experimental and FEM results in Figure 2 have the same trend with a high degree of agreement, and the deviations are all within 10%. Therefore, the FEMs in Section 2.2 can predict the mechanical properties of SPAA.

#### 4.2. Contact State and Mechanical Properties

_{t}), normal adhesion force (F

_{n}) of the SPAA, and contact areas of four APs (S

_{AP1}~S

_{AP4}) versus the time. During the preload, only AP1 contacts the cylinder and generates a normal squeezing force of approximately 1 N. During the adhesion, AP2, AP3, and AP4 are in contact with the arc surface orderly as the air pressure increases. Meantime, the SPAA exerts a small tangential adhesion force to the cylinder away from the root. It is mainly because the inextensible layer of the SPAA inevitably generates a slight tensile deformation during the expansion of the SPAA. During the peeling, the SPAA does not peel off the cylinder once, but generates two peeling processes. In the first peeling, the F

_{t}and F

_{n}go up and S

_{AP1}~S

_{AP3}go down synchronously. When the F

_{t}and F

_{n}reach the peak, they fall back rapidly, S

_{AP1}~S

_{AP3}drop to zero at the same time. In the second peeling, the F

_{t}and F

_{n}increase again but at a slower rate, together with the slight decrease in the S

_{AP4}. When the F

_{t}and F

_{n}reach the peak, F

_{t}, F

_{n}, and S

_{AP4}drop to zero at the same time.

#### 4.3. Adhesion-Peeling Performance on Large Cylinders

_{AP1}~S

_{AP4}) during the adhesion stage. Obviously, the S

_{AP4}increases the most with an increase in the rib height. As the rib height increases from 4 to 8 mm, the contact ratio of the end region increases from 40% to 70%, indicating that a higher rib of the SPAA benefits a more sufficient contact. A low standard deviation means that the contact area state of each AP is little affected by the curvature of the cylinder. We also found that AP3 has the lowest contact area with a 15% contact ratio, indicating that the adhesion performance of the area near AP3 is also lower than that of other APs.

_{t}and F

_{n}in the first and second peelings of three SPAAs with different rib heights, and defined as F

_{t_peak1}, F

_{n_peak1}, F

_{t_peak2}, and F

_{n_peak2}, respectively (Figure 5b). It is evident that with the rib height of the SPAA increasing, the peak adhesive forces show an upward trend. As the rib height increases from 4 mm to 8 mm, the forces F

_{n_peak1}and F

_{n_peak2}stabilize in the range of 7~10 N and 2~6 N, respectively, and increase slowly. The tangential adhesion force also increases with the rib height increasing, but it is significantly affected by the peeling angle θ

_{p}. Specifically, the force F

_{t_peak2}at θ

_{p}= 30° increases from 6 N to 10~12 N, the most evident increase. The above results mean that the increase in the rib height is conducive to a more outstanding adhesion performance of the SPAA on large cylinders, and the effect is more significant when at a smaller peeling angle. Further, the average increases of F

_{n_peak1}, F

_{t_peak1}, F

_{n_peak2}, and F

_{t_peak2}are 20%, 31%, 102%, and 62%, respectively. The increase in the peak adhesion forces in the second peeling is 2~5 times that in the first peeling. Since the F

_{t_peak2}and F

_{n_peak2}are generated by the peeling off of the AP4 (Section 4.2), it can be verified that the increase in the rib height can significantly improve the adhesion performance of the SPAA’s end contact region.

_{p}on the peak tangential adhesion force is more significant than that of the peak normal adhesion force. As the θ

_{p}decreases from 90° to 30°, F

_{t_peak1}increases from 2~4 N to 8~12 N, with an average increase of 290%; the F

_{t_peak2}increases from approximately 4 N to 6~12 N, with an average increase of 95%; while the increase in peak normal adhesion force is very small, only 2%. When peeling at 90°, the F

_{t_peak1}is only approximately 10%~30% of the F

_{n_peak1}, and the F

_{t_peak2}is equivalent to the F

_{n_peak2}. However, when peeling at 30°, F

_{t_peak1}is basically the same as the F

_{n_peak1}, and the F

_{t_peak2}is 2~3 times of the F

_{n_peak2}. In general, the decrease in θ

_{p}is beneficial to a more excellent tangential adhesion performance of the SPAA on large cylinders, and the effect is more significant when the SPAA has a larger flexural rigidity. In addition, SPAAs with different bending rigidities show a very stable normal adhesion performance when peeling at different angles on large cylinders. Cylinder curvature has little effect on the adhesion performance of the SPAA. As the curvature decreases, the overall adhesion force decreases slightly, indicating that the SPAA can generate a stable adhesion performance when adhering to cylinders with large curvatures.

## 5. Discussion

#### 5.1. Comparison between the SPAA and Other Adhesive Units

#### 5.2. Optimization for High Contact Ratio

^{3}. Compared to ordinary steel, such as aluminum alloy, carbon plates are lighter in weight and higher in bending strength. Finally, four carbon plates (3 mm * 1.5 mm * 30 mm) were selected and embedded inside the inextensible layer (Figure 8e). The optimized SPAA has a hierarchical structure with variable rigidity, which prevents the SPAA’s Aps from expanding radially into an arc under positive pressure while maintaining the SPAA’s axially flexible bending performance. The FEM results in Figure 8c,f show that the contact area of the AP2 has changed from an intermediate distribution (homogeneous SPAA) to a balanced radial distribution along the AP2 (SPAA with variable rigidity), and the contact area has increased from 70 to 210 mm

^{2}, an increase of 200% (Figure 8g).

_{4}) is slightly greater than that of PN3 (C

_{3}), resulting in an over-bending angle of PN4 with respect to PN3, i.e., $\left(l{C}_{4}-l{C}_{3}\right)/2$ (Figure 8h). This over-bending angle directly causes AP4 to squeeze tightly with the cylinder, while AP3 cannot contact the cylinder.

_{AP1}and S

_{AP2}are almost unaffected by ${\theta}_{\mathrm{T}}$, while S

_{AP3}and S

_{AP4}rise rapidly to a peak as ${\theta}_{\mathrm{T}}$ increases from 0° to 3°, after which S

_{AP3}is maintained. However, as ${\theta}_{\mathrm{T}}$ continues to raise, S

_{AP4}begins to fall, and the pressure P also reaches 100 kPa at this point. Since the excessive ${\theta}_{\mathrm{T}}$ requires higher air pressure to drive larger bending deformation of the SPAA to fit AP4 to the cylinder, the ${\theta}_{\mathrm{T}}$ at the point S

_{AP4}begins to fall increases as the radius of the cylinder increases. The results of the SPAA on all three cylinders show that a good contact state can be achieved with ${\theta}_{\mathrm{T}}$ between 3° and 6°, and the contact ratio of four APs is between 70% and 100%, which is a noticeable optimization compared to the SPAA with untitled AP4. Too low a ${\theta}_{\mathrm{T}}$ would result in inadequate contact of AP3, and too high a ${\theta}_{\mathrm{T}}$ would increase the air pressure to reach a steady-adhesion state. The final determination is ${\theta}_{\mathrm{T}}$ = 5°, at which the overall contact ratio of the SPAA reaches over 80%.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) The finite element model of the bending curvature of the SPAA. (

**b**) The curvature of four PNs when the SPAA under pressure drive. (

**c**) The bending curvature of three SPAAs with different rib heights under pressure drive. (

**d**) The finite element model of the flexural rigidity of the SPAA. (

**e**) The flexural rigidity of three SPAAs with different rib heights under pressure drive.

**Figure 3.**Experimental setup. (

**a**) The synchronous testing platform. (

**b**) Assembly layout of the SPAA, the six-dimensional force sensor and the pressure regulator. (

**c**) Three acrylic semi-cylinders with radii of 100 mm, 150 mm and 200 mm. (

**d**) Three SPAA samples prepared by 3D printing and their structural parameters.

**Figure 4.**(

**a**) The bending state of three SPAAs with rib heights of 4 mm, 6 mm, and 8 mm at 0 kPa, 50 kPa, and 100 kPa, respectively. (

**b**) The experimental results of the bending curvature of three SPAAs with different rib heights under pressure drive. (

**c**) The test flow of SPAA’s flexural rigidity. (

**d**) The experimental results of the flexural rigidity of three SPAAs with different rib heights under pressure drive.

**Figure 5.**(

**a**) The flow of the adhesion-peeling performance test of the SPAA. (

**b**) A typical curve of tangential adhesion force (F

_{t}), normal adhesion force (F

_{n}) of the SPAA, and contact areas of four APs (S

_{AP1}~S

_{AP4}) versus the time.

**Figure 6.**The experimental results of the contact area of four APs (S

_{AP1}~S

_{AP4}) during the adhesion stage of three SPAAs.

**Figure 7.**The peak forces of F

_{t}and F

_{n}in the first and second peelings of three SPAAs with different “rib” heights peeling off from cylinders with radii of 100 mm, 150 mm, 200 mm at peeling angles of 90°, 60°, 30°.

**Figure 8.**(

**a**) The contact state of the SPAA during adhesion. (

**b**) The finite element model describing the contact of the SPAA. (

**c**) The simulation results of the contact areas of four Aps of the original SPAA with homogeneous structure. (

**d**) The simulated cross-section of the SPAA in contact with the cylinder. (

**e**) The optimized SPAA with a rigid–flexible coupled hierarchical structure. (

**f**) The simulation results of the contact areas of four APs of the optimized SPAA with variable rigidity. (

**g**) The increase. (

**h**) Illustration of the over-bending angle of PN4 with respect to PN3. (

**i**) The optimized SPAA with a differentiated AP structure. (

**j**) The contact area and air pressure of each AP versus ${\theta}_{\mathrm{T}}$ (0°~12°) when the SPAA with an 8 mm rib height reaches the steady adhesion.

**Table 1.**Comparison of the adhesion performance of the existing adhesive grippers consisting of soft adhesive actuators.

Soft Adhesive Gripper | Actuating Technology | Single Unit Size (mm) | Detach Force (N)/ Detach Angle (°)/ Radium (mm) |
---|---|---|---|

[13] | Fluidic–elastic actuator (rubber) | 60 ∗ 20 | 18/≈60/75 9.5/≈75/101.5 |

[14] | Fluidic–elastic actuator (fabric) | 35 ∗ 24 | 4.3/90/14 |

[5] | Under-actuated | ≈35 ∗ 100 | 5.5/≈45/100 12/≈55/150 |

[34] | Under-actuated | 50 ∗ 32 | 5.5/≈45/100 3.5/≈60/200 1.8/≈80/400 |

[35] | Shape memory alloy-actuated | 100 ∗ 15 | 10/0/62.5 |

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**MDPI and ACS Style**

Wang, L.; Jiang, Q.; Weng, Z.; Yuan, Q.; Wang, Z.
Effects of Flexural Rigidity on Soft Actuators via Adhering to Large Cylinders. *Actuators* **2022**, *11*, 286.
https://doi.org/10.3390/act11100286

**AMA Style**

Wang L, Jiang Q, Weng Z, Yuan Q, Wang Z.
Effects of Flexural Rigidity on Soft Actuators via Adhering to Large Cylinders. *Actuators*. 2022; 11(10):286.
https://doi.org/10.3390/act11100286

**Chicago/Turabian Style**

Wang, Liuwei, Qijun Jiang, Zhiyuan Weng, Qingsong Yuan, and Zhouyi Wang.
2022. "Effects of Flexural Rigidity on Soft Actuators via Adhering to Large Cylinders" *Actuators* 11, no. 10: 286.
https://doi.org/10.3390/act11100286