# An Optimized Design of the Soft Bellow Actuator Based on the Box–Behnken Response Surface Design

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- To study the parameters that influence the deformation response of the soft bellow actuator;
- To determine the optimized model of the soft bellow actuator and analyze the significant parameters using ANOVA;
- To propose a regression model to predict the deformation and stress response for this specific design of the soft bellow actuator;
- To validate the optimized design through experimental testing, assessing the deformation response and grasping performance using pulling force testing.

## 2. Materials and Methods

#### 2.1. Workflow

#### 2.2. A Mathematical Model of Hyper-Elastic Materials

_{1}and I

_{2}are the deformation invariants, whereas C

_{ij}are material-specific constants. These deformation invariants can be expressed as follows:

_{ij}parameter as follows:

#### 2.3. Design and Statistical Analysis of Experiments

_{i}and x

_{j}are the independence parameters; β

_{0}is the intercept coefficient; and β

_{i}, β

_{ii}, and β

_{ij}are the regression coefficients for linear, quadratic, and interaction variables, respectively. The experimental miscalculation (γ) is assumed to be zero. This regression model was used to estimate the factor responses. The fitness of the polynomial regression is defined by the coefficient of determination (R

^{2}). The significance of the factors is evaluated using ANOVA, employing the f-value at a specified p-value threshold. The optimal conditions for each factor are estimated through three-dimensional (3D) response surface plots and contour plots. Subsequently, the optimized model of the soft bellow actuator was validated by conducting experiments to verify the response.

## 3. Finite Element Analysis of Soft Bellow Design

#### 3.1. Geometric Model of the Soft Bellow Actuator

_{1}) is 40 mm, the height of the soft actuator (H

_{1}) is 35 mm, and the length from the fixed base (H

_{2}) is 5 mm. The height of the bellows layer (h) is determined by the number of bellows (N) and the wall thickness (t). The relationship between these parameters can be expressed as follows:

#### 3.2. Simulation Analysis

_{10}= 1.192 × 10

^{−1}MPa, C

_{01}= 3.464 × 10

^{−9}MPa, and C

_{11}= 1.484 × 10

^{−2}MPa, with an error 3.370 × 10

^{−4}. To optimize the design of the soft bellow gripper, the deformation and stress at the center of the model’s upper surface were analyzed. These response parameters were crucial for the DOE using the Box–Behnken method, allowing for the design of the bellow soft gripper with acceptable levels of deformation and stress.

## 4. Box–Behnken Design

#### Analysis of Variance

^{2}and adj-R

^{2}values for this regression model are 0.998 and 0.9943, respectively. These values exceed 0.7, indicating a significant correlation between the displacement response and the independent variables. R

^{2}values closer to 1 suggest that the regression model is well-suited to the actual response.

^{2}and adj-R

^{2}values of 0.9954 and 0.9871, respectively. This indicates that the model accurately predicts the stress of the actuator.

## 5. Manufacturing and Assembly

^{3}and weighs 17.89 g, considering the density of Dragon Skin 30 at 1.08 g/cm

^{3}. Figure 9 shows the four steps involved in the casting process. The model’s mold is divided into five pieces for easy disassembly and was created using a 3D printer, specifically the Flashforge Adventurer 3 (Zhejiang Flashforge 3D Technology Co., Ltd., Jinhua, China).

## 6. Experiments and Results

#### 6.1. Validation of the Deformation Response

#### 6.2. Pulling force Experiment

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Simulation result of deformation response under 50 kPa pressure with different wall thicknesses of (

**a**) 3 mm, (

**b**) 4.5 mm, and (

**c**) 6 mm.

**Figure 6.**Relationship between deformation and air pressure of simulation results (

**a**) under different wall thicknesses, (

**b**) under different numbers of bellows, (

**c**) under different divided ratios, and (

**d**) under different heights of the actuator.

**Figure 7.**Response surface and contour diagrams of the soft bellow: (

**a**) under constant the divided ratio of radians, (

**b**) under constant the number of bellows, and (

**c**) under constant the wall thickness.

**Figure 9.**Fabrication of a Soft Bellow Actuator: (

**a**) 3D-printed molds, (

**b**) silicone preparation, (

**c**) casting process, and (

**d**) final assembly.

**Figure 10.**Assembly of the rigid frame and the soft bellow actuator (

**a**) in the 3D model and (

**b**) in the real model.

**Figure 12.**The deformation response at different air pressure levels: (

**a**) at 2 kPa, (

**b**) at 30 kPa, and (

**c**) at 50 kPa.

**Figure 15.**Responses of the pulled gripper at different air pressure levels of (

**a**) 20 kPa, (

**b**) 25 kPa, (

**c**) 30 kPa, (

**d**) 35 kPa, and (

**e**) 40 kPa.

Factor | Name | Low | Median | High |
---|---|---|---|---|

A | Wall thickness | 3 | 4.5 | 6 |

B | Number of bellows | 3 | 6 | 9 |

C | Divided ratio of radians | 2 | 3 | 4 |

Standard Order | Running Order | Point Type | Blocks | A | B | C | D | E |
---|---|---|---|---|---|---|---|---|

1 | 1 | 2 | 1 | 3 | 3 | 3 | 18.06 | 290.8 |

2 | 2 | 2 | 1 | 6 | 3 | 3 | 6.49 | 243.9 |

3 | 3 | 2 | 1 | 3 | 9 | 3 | 13.28 | 287.9 |

4 | 4 | 2 | 1 | 6 | 9 | 3 | 5.19 | 234.1 |

5 | 5 | 2 | 1 | 3 | 6 | 2 | 19.02 | 288.2 |

6 | 6 | 2 | 1 | 6 | 6 | 2 | 6.37 | 250.1 |

7 | 7 | 2 | 1 | 3 | 6 | 4 | 13.19 | 289.7 |

8 | 8 | 2 | 1 | 6 | 6 | 4 | 5.14 | 232.0 |

9 | 9 | 2 | 1 | 4.5 | 3 | 2 | 16.07 | 294.6 |

10 | 10 | 2 | 1 | 4.5 | 9 | 2 | 9.52 | 294.3 |

11 | 11 | 2 | 1 | 4.5 | 3 | 4 | 9.50 | 288.6 |

12 | 12 | 2 | 1 | 4.5 | 9 | 4 | 8.27 | 290.8 |

13 | 13 | 0 | 1 | 4.5 | 6 | 3 | 9.12 | 291.8 |

14 | 14 | 0 | 1 | 4.5 | 6 | 3 | 9.12 | 291.8 |

15 | 15 | 0 | 1 | 4.5 | 6 | 3 | 9.12 | 291.8 |

Source | DF | Adj SS | Adj MS | F-Value | p-Value |
---|---|---|---|---|---|

Model | 9 | 277.84 | 30.87 | 273.89 | <0.0001 |

Linear | |||||

A | 1 | 203.59 | 203.59 | 1806.27 | <0.0001 |

B | 1 | 23.97 | 23.97 | 212.64 | <0.0001 |

C | 1 | 27.68 | 27.68 | 245.6 | <0.0001 |

Square | |||||

A× | 1 | 2.76 | 2.76 | 24.49 | 0.0043 |

B×B | 1 | 2.22 | 2.22 | 19.66 | 0.0068 |

C×C | 1 | 3.31 | 3.31 | 29.38 | 0.0029 |

2-way Interaction | |||||

A×B | 1 | 3.05 | 3.05 | 27.02 | 0.0035 |

A×C | 1 | 5.3 | 5.3 | 47.01 | 0.001 |

B×C | 1 | 7.06 | 7.06 | 62.65 | 0.0005 |

Error | 5 | 0.5636 | 0.1127 | ||

Total | 14 | 278.4 | |||

R^{2} = 0.998, adj-R^{2} = 0.9943, C.V.% = 3.2, and Adeq Precision = 51.55 |

Source | DF | Adj SS | Adj MS | F-Value | p-Value |
---|---|---|---|---|---|

Model | 9 | 7830.61 | 870.07 | 119.83 | <0.0001 |

Linear | |||||

A | 1 | 4827.23 | 4827.23 | 664.82 | <0.0001 |

B | 1 | 14.86 | 14.86 | 2.05 | 0.212 |

C | 1 | 84.18 | 84.18 | 11.59 | 0.0191 |

Square | |||||

A×A | 1 | 2757.63 | 2757.63 | 379.79 | <0.0001 |

B×B | 1 | 0.2251 | 0.2251 | 0.031 | 0.8671 |

C×C | 1 | 1.12 | 1.12 | 0.1541 | 0.7108 |

2-way Interaction | |||||

A×B | 1 | 11.73 | 11.73 | 1.62 | 0.2597 |

A×C | 1 | 96.82 | 96.82 | 13.33 | 0.0147 |

B×C | 1 | 1.67 | 1.67 | 0.2299 | 0.6518 |

Error | 5 | 36.3 | 7.26 | ||

Total | 14 | 7866.92 | |||

R^{2} = 0.9954 and adj-R^{2} = 0.9871, C.V.% = 0.9716, and Adeq Precision = 29.572 |

Pressure (kPa) | Deformation (mm) | Strain (%) | % Error of a Deformation | ||
---|---|---|---|---|---|

Experiment (Average) | Simulation | Experiment (Average) | Simulation | ||

5 | 1.22 | 5.37 | 0.035 | 0.153 | −77.29 |

10 | 4.452 | 7.81 | 0.127 | 0.223 | −42.99 |

15 | 6.672 | 9.66 | 0.191 | 0.276 | −30.92 |

20 | 8.864 | 11.26 | 0.253 | 0.322 | −21.29 |

25 | 10.224 | 12.73 | 0.292 | 0.364 | −19.68 |

30 | 11.76 | 14.11 | 0.336 | 0.403 | −16.63 |

35 | 13.124 | 15.41 | 0.375 | 0.440 | −14.85 |

40 | 14.544 | 16.66 | 0.416 | 0.476 | −12.72 |

45 | 15.74 | 17.86 | 0.450 | 0.510 | −11.89 |

50 | 16.84 | 19.02 | 0.481 | 0.543 | −11.46 |

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## Share and Cite

**MDPI and ACS Style**

Auysakul, J.; Booranawong, A.; Vittayaphadung, N.; Smithmaitrie, P.
An Optimized Design of the Soft Bellow Actuator Based on the Box–Behnken Response Surface Design. *Actuators* **2023**, *12*, 300.
https://doi.org/10.3390/act12070300

**AMA Style**

Auysakul J, Booranawong A, Vittayaphadung N, Smithmaitrie P.
An Optimized Design of the Soft Bellow Actuator Based on the Box–Behnken Response Surface Design. *Actuators*. 2023; 12(7):300.
https://doi.org/10.3390/act12070300

**Chicago/Turabian Style**

Auysakul, Jutamanee, Apidet Booranawong, Nitipan Vittayaphadung, and Pruittikorn Smithmaitrie.
2023. "An Optimized Design of the Soft Bellow Actuator Based on the Box–Behnken Response Surface Design" *Actuators* 12, no. 7: 300.
https://doi.org/10.3390/act12070300