# Design Methodology for a Novel Bending Pneumatic Soft Actuator for Kinematically Mirroring the Shape of Objects

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Development of the Numerical Model

#### 2.1. The Conceptual Idea of the Actuator

#### 2.2. The Numerical Model

^{®}Matlab environment. A graphical user interface (GUI), shown in Figure 1b, was created in order to input the following parameters:

- -
- Fi: the length of the i-th phalanx, measured in mm.
- -
- openFi: the axial length of the i-th cut, measured in mm.
- -
- alfa_center: in correspondence of the i-th cut region (i.e., i-th joint), the angular position of the center of the uncut sector of the gauze, measured in °.
- -
- alfa_closed: in correspondence of the i-th cut region, the angular extension of the uncut sector of the gauze, measured in °. It is always considered symmetrical with respect to the alfa_center value.
- -
- Rext: the external radius of the inner hyper-elastic tube, measured in mm.
- -
- Root: the length of the metacarpal or, more generally, of the initial part of the actuator attached to a support, measured in mm.
- -
- th: the thickness of the inner hyper-elastic tube, measured in mm.
- -
- Pressure: the maximum value of the compressed air pressure in the actuator, measured in bar.

_{10}= 0.0694 MPa and C

_{01}= 0.0628 MPa, ν = 0.46) of the Mooney–Rivlin formulation; the nondeformable closed end is modelled as a linear isotropic material, with the mechanical properties of aluminum (E = 70000 MPa; ν = 0.33); the gauze material is modelled as linear isotropic, with the mechanical properties of polyamide (E = 2000 MPa, ν = 0.10). A fixed constraint is applied at the end in correspondence of the Root. Pressure is applied to the internal faces of the inner elements of the tube. The gravity effect was neglected. The Newton–Raphson method was implemented to perform the nonlinear analysis: pressure was applied from zero to the Pressure value, according to a time step automatically chosen by the FEM code.

_{1}≤ 75°,

_{2}≤ 100°,

_{3}≤ 65°.

#### 2.3. The Experimental Prototype

#### 2.4. The Experimental Validation of the Numerical Model

_{1}, θ

_{2}and θ

_{3}, depending on the pressure value. A test bed, made of aluminum commercial profiles (30 × 30 Rexroth Bosch Group) mounted to realize a portal structure was realized; a manometer (Nuova Fima DN-100), with a resolution of 0.05 bar, was adopted for the pressure measurement; a precise pressure regulator (SMC IR-1000) completed the test bed. The end of the actuator with the aluminium plug was fixed, with a couple of nuts, to a bracket rigidly fixed to the test bed. The actuator sticks out from the test bed. A 4 mm external diameter polyurethane tube (SMC Corporation) was adopted for the air inlet/outlet. Tests were carried out starting from zero pressure, with an increase of 0.10 bar, until the value of 2.30 bar, corresponding to the closure of the finger, over the conditions expressed in (1)–(3). About 5 s was taken to manually adjust the pressure regulator in order to reach the next pressure step. When each pressure step was reached, about 10 s was spent to check if residual deformations, due to the viscoelastic behavior of the silicone rubber, occurred. Then, a picture of the actuator was taken. No residual deformations were ever recorded. The time between the reaching of the two consecutive pressure steps was about 30 s.

_{2}. At the same pressure value, the model provides for greater angles until 1.50 bar; then, the experimental prototype exceeds. The latter result is well described by the comparison of the trajectories followed by the midpoints of each experimental and numerical phalanx, as shown in Figure 4.

^{2}, close to 1, confirmed the validation of the model. Moreover, the experimental work tested the functionality of such a kind of actuator. For pressure values lower than 2.00 bar, the angles of the joint satisfied the relations (1)–(3); for values higher than 2.00 bar, the tube showed some bulges in correspondence of the joints.

## 3. The Predictive Formula

_{i}) can be expressed as:

_{i}= θ

_{i}(D, s, α

_{i}, openFi, P),

_{i}= θ

_{i}(α

_{i}, openFi, P),

_{i}, while an increase in α

_{i}provides for a decrease in θ

_{i}.

#### 3.1. Significant Range of α and Preliminary Consideration about θ vs. OpenF

^{2}value is close to 1 (for sake of clarity, some equations have not been reported). As expected, the increase in α provides for a decrease in θ.

_{j}, P

_{0.2})∙openF

_{l}+ c(α

_{j}, P

_{0.2})

#### 3.2. Achievement of the Predictive Formula

_{j}, P

_{k})∙openF

_{l}+ c(α

_{j}, P

_{k})

_{j}, P

_{k}) and c(α

_{j}, P

_{k}), the methodology proposed in [37] was applied: for each α

_{j}, m(α

_{j}, P

_{k}) and c(α

_{j}, P

_{k}), values were collected and plotted as a function of P, as shown in Figure 7. Then, the shapes and the equations of the regression lines were found. Due to the value of R

^{2}, the behavior of m(α

_{j}, P

_{k}) can be well described by a parabolic curve; on the contrary, the behavior of c(α

_{j}, P

_{k}) can be described by a discrete approximation of a line. The expression (7) can be improved as:

_{1}(α

_{j})∙P

^{2}+ ã

_{2}(α

_{j})∙P]∙openF

_{l}+ [ã

_{3}(α

_{j})]∙P,

_{1}(α

_{j}) and ã

_{2}(α

_{j}) represent the set of the coefficients of the equation of the parabolic curves of Figure 7a and ã

_{3}(α

_{j}) represents the set of the gradients of the lines of Figure 7b.

_{1}(α

_{j}), ã

_{2}(α

_{j}) and ã

_{3}(α

_{j}) coefficients have been plotted as a function of α, as shown in Figure 7c: two parabolic curves and one line better describe their behaviors, respectively. The coefficients of the equations of the regression curves were adopted to finalize the formula.

_{11}∙α

^{2}+c

_{12}∙α+c

_{13})∙P

^{2}+ (c

_{21}∙α

^{2}+c

_{22}∙α+c

_{23})∙P]∙openF

_{l}+ [(c

_{31}∙α+c

_{32}) ∙P]

#### 3.3. Validation of the Formula

## 4. The Design Methodology: A Case Study

- The import of the geometry/picture of the object to be grasped in a CAD environment (see the picture of the light bulb).
- The design of the useful profile of the object to be grasped (see the black inner profile on the left side of the light bulb).
- The design of the external profile of the actuator as an external offset of the inner profile, according to the value of the diameter of the tube (see the blue profile with an offset of 20 mm).
- The discretization of the wanted shape of the actuator in a series of joints (indicated by the numbers in the red circles). Each joint is made of a bending sector (see the dashed red areas, BS
_{i}) among two nondeformable sectors (see the dashed blue areas, NS_{i}) that will be totally wrapped by the gauze.Generally, it is better to increase the number of joints to better follow the desired profile. The centers of the bending sectors must coincide with the center of the curvature of the profile of the object. The length of each bending sector is arbitrary; nevertheless, the greater the length, the greater the effect of the polygonal chain, and a non-optimal fitting could occur. - Measurements of the following entities:
- -
- θ of the i-th joint (see the red angular dimensions): it is the bending angle measured between two lines of discretization.
- -
- openF of the i-th joint (see the red linear dimensions): it is measured as the linear dimension of the bending sector on the inner profile. After the discretization, the openF value must be approximated by excess to become the closest integer and multiple of two (due to the dimensions of the adopted gauze).
- -
- The length of the i-th nondeformable sector between two adjacent joints: it is the linear dimension (see the blue linear dimensions) measured between the intersection points of the curved axis of the wanted shape with the perpendicular lines starting from the extremities of the nondeformable sector placed on the inner profile.

- Achievement of the length of Fi: as shown in Figure 1b, it is the linear dimension computed asFi = openF
_{i}/2 + length(NS_{i}) + openF_{i+1}/2 - Collecting the values of F, openF, alfa_center and θ for each joint.
- The application of the formula to derive the values of α
_{i}and the maximum operating pressure value.

_{i}values are achieved by:

- The recognition of the maximum operating pressure value: entry into the table by the column of openF
_{i}corresponding to the highest value of θ_{i}resulting from Steps 1–7; from the row corresponding to the highest value of the pressure towards rows with lower values of it, for each row, move from the left to the right until the current value of θ_{i}is equal to the wanted one or to a next higher available value of it; the row including the cell with such a value provides the maximum operating pressure value (to be read from the left column of Table 4), while the column including it provides the α_{i}value (to be read from the lowest row of Table 4). In the case reported here, the highest value of θ_{i}is 45° and the corresponding value of openF_{i}is 10 mm. Then, in the column of openF_{i}= 10, the value of θ_{i}nearest to the wanted one is 45.3°, in the black thick line cell. From this cell, moving leftward, the maximum operating pressure value is equal to 1.90 bar (in the green thick line cell); moving downward, the α_{i}value is equal to 132° (in the blue thick line cell). - The recognition of the values of the remaining parameters: in the row of the recognized maximum operating pressure value, for the i-th joint, seek the entry in the column with the wanted openF
_{i}and find the closest and highest value of the corresponding θ_{i}; then, moving downward, the value of α_{i}is recognized as previously described.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The numerical model: (

**a**) a three-phalanx finite element model of the actuator; (

**b**) a view of the developed graphical user interface (GUI); (

**c**) the expected kinematic behavior of the actuator; (

**d**) a schematic of the geometry of the cuts for the bending of a phalanx.

**Figure 2.**The experimental prototype: (

**a**,

**b**): the mold for the tube; (

**c**,

**d**): welding steps for the gauze; (

**e**) the aluminum plug; (

**f**) an overall view; (

**g**) a detail of the cuts in the gauze.

**Figure 3.**The validation of the finite element method (FEM) model: (

**a**) measurements of experimental and numerical bending angles; (

**b**) comparison between experimental and numerical bending angles; (

**c**) results of the correlation and regression analyses between experimental and numerical bending angles.

**Figure 7.**Steps for the achievement of the formula: (

**a**) m vs. P; (

**b**) c vs. P; (

**c**) a1, a2 and a3 vs. α.

**Figure 8.**Comparison between numerical and formulated θ for different values of pressure (0.2, 0.6, 1.0, 1.6 and 2.0 bar, from the higher to the lower row, respectively) and for different values of α (84°, 120° and 156°, left, centre and right column, respectively).

**Figure 10.**Evaluation of the designed actuator: numerical model and lateral and front views of the experimental prototype.

MC Length (mm) | PP Length (mm) | IP Length (mm) | DP Length (mm) | Soft Tissue Length of the Tip of DP (mm) |
---|---|---|---|---|

68.12 ± 6.27 | 39.78 ± 4.94 | 22.38 ± 2.51 | 15.82 ± 2.26 | 3.84 ± 0.59 |

F1(mm) | F2(mm) | F3 (mm) | openF1(mm) | openF2(mm) | openF3(mm) |

40 | 22 | 20 | 14 | 16 | 12 |

α_center (°) | α_closed (°) | Rext (mm) | Root(mm) | th(mm) | Pressure(bar) |

270 | 90 | 10 | 10 | 2 | 2.30 |

c_{11} | c_{12} | c_{13} | c_{21} | c_{22} | c_{23} | c_{31} | c_{32} |
---|---|---|---|---|---|---|---|

−0.000166 | 0.031915 | −0.856229 | −0.000009 | −0.000689 | 0.991593 | −0.007112 | 4.051633 |

OpenFi (mm) | 6 | 8 | 10 | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Pressure (bar) | θ (°) | θ (°) | θ (°) | |||||||||||||||||||||

0.20 | 1.9 | 1.8 | 1.8 | 1.7 | 1.6 | 1.5 | 1.4 | 1.3 | 2.3 | 2.2 | 2.2 | 2.1 | 2.0 | 1.8 | 1.7 | 1.5 | 2.7 | 2.6 | 2.5 | 2.4 | 2.3 | 2.1 | 2.0 | 1.7 |

0.40 | 4.1 | 4.0 | 3.9 | 3.7 | 3.5 | 3.2 | 2.9 | 2.5 | 5.0 | 4.9 | 4.8 | 4.5 | 4.2 | 3.9 | 3.4 | 2.9 | 5.9 | 5.8 | 5.6 | 5.3 | 5.0 | 4.5 | 4.0 | 3.3 |

0.60 | 6.6 | 6.5 | 6.3 | 6.0 | 5.5 | 5.0 | 4.3 | 3.6 | 8.1 | 8.0 | 7.8 | 7.3 | 6.8 | 6.1 | 5.2 | 4.2 | 9.6 | 9.5 | 9.2 | 8.7 | 8.0 | 7.1 | 6.1 | 4.8 |

0.80 | 9.4 | 9.3 | 9.0 | 8.5 | 7.8 | 7.0 | 5.9 | 4.6 | 11.7 | 11.6 | 11.2 | 10.5 | 9.6 | 8.5 | 7.0 | 5.3 | 13.9 | 13.8 | 13.3 | 12.5 | 11.4 | 10.0 | 8.2 | 6.1 |

1.00 | 12.6 | 12.5 | 12.1 | 11.4 | 10.4 | 9.0 | 7.4 | 5.5 | 15.6 | 15.5 | 15.0 | 14.1 | 12.8 | 11.1 | 8.9 | 6.4 | 18.7 | 18.6 | 17.9 | 16.8 | 15.2 | 13.1 | 10.4 | 7.3 |

1.20 | 16.1 | 16.0 | 15.4 | 14.5 | 13.1 | 11.3 | 9.0 | 6.4 | 20.0 | 19.9 | 19.3 | 18.0 | 16.2 | 13.8 | 10.9 | 7.3 | 24.0 | 23.9 | 23.1 | 21.6 | 19.3 | 16.4 | 12.7 | 8.3 |

1.30 | 17.9 | 17.8 | 17.2 | 16.1 | 14.6 | 12.5 | 9.9 | 6.7 | 22.4 | 22.3 | 21.6 | 20.1 | 18.1 | 15.3 | 11.9 | 7.8 | 26.8 | 26.8 | 25.9 | 24.1 | 21.6 | 18.1 | 13.9 | 8.8 |

1.40 | 19.8 | 19.8 | 19.1 | 17.9 | 16.1 | 13.7 | 10.7 | 7.1 | 24.8 | 24.8 | 24.0 | 22.3 | 20.0 | 16.8 | 12.9 | 8.1 | 29.8 | 29.8 | 28.8 | 26.8 | 23.9 | 19.9 | 15.1 | 9.2 |

1.50 | 21.8 | 21.8 | 21.1 | 19.7 | 17.6 | 14.9 | 11.5 | 7.5 | 27.4 | 27.4 | 26.5 | 24.6 | 22.0 | 18.4 | 13.9 | 8.5 | 32.9 | 32.9 | 31.8 | 29.6 | 26.3 | 21.8 | 16.3 | 9.6 |

1.60 | 23.9 | 23.9 | 23.1 | 21.6 | 19.3 | 16.2 | 12.4 | 7.8 | 30.0 | 30.1 | 29.1 | 27.0 | 24.0 | 20.0 | 14.9 | 8.9 | 36.2 | 36.2 | 35.0 | 32.5 | 28.8 | 23.7 | 17.5 | 9.9 |

1.70 | 26.1 | 26.1 | 25.2 | 23.5 | 20.9 | 17.5 | 13.2 | 8.1 | 32.8 | 32.9 | 31.8 | 29.5 | 26.1 | 21.6 | 16.0 | 9.2 | 39.5 | 39.6 | 38.3 | 35.5 | 31.4 | 25.7 | 18.7 | 10.2 |

1.80 | 28.3 | 28.3 | 27.4 | 25.5 | 22.7 | 18.9 | 14.1 | 8.4 | 35.7 | 35.8 | 34.6 | 32.1 | 28.3 | 23.3 | 17.0 | 9.4 | 43.0 | 43.2 | 41.7 | 38.7 | 34.0 | 27.8 | 19.9 | 10.5 |

1.90 | 30.6 | 30.7 | 29.7 | 27.6 | 24.4 | 20.2 | 15.0 | 8.6 | 38.7 | 38.8 | 37.5 | 34.8 | 30.6 | 25.1 | 18.1 | 9.7 | 46.7 | 46.9 | 45.3 | 41.9 | 36.8 | 29.9 | 21.2 | 10.8 |

2.00 | 33.0 | 33.1 | 32.0 | 29.7 | 26.3 | 21.7 | 15.9 | 8.9 | 41.7 | 41.9 | 40.5 | 37.5 | 33.0 | 26.9 | 19.2 | 9.9 | 50.4 | 50.7 | 49.0 | 44.9 | 39.7 | 32.1 | 22.5 | 11.0 |

alfa_closed [°] | 84 | 96 | 108 | 120 | 132 | 144 | 156 | 168 | 84 | 96 | 108 | 120 | 132 | 144 | 156 | 168 | 84 | 96 | 108 | 120 | 132 | 144 | 156 | 168 |

Joint Number | F (mm) | OpenF (mm) | α_Closed (°) | α _Center (°) | θ (°) | Pmax (bar) |
---|---|---|---|---|---|---|

1 | 19 | 8 | 132 | 270 | 30 | 1.90 |

2 | 19 | 8 | 132 | 270 | 30 | |

3 | 19 | 8 | 132 | 270 | 30 | |

4 | 12 | 8 | 144 | 270 | 25 | |

5 | 12 | 6 | 144 | 270 | 20 | |

6 | 8 | 10 | 108 | 90 | 45 |

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Antonelli, M.G.; Beomonte Zobel, P.; D’Ambrogio, W.; Durante, F. Design Methodology for a Novel Bending Pneumatic Soft Actuator for Kinematically Mirroring the Shape of Objects. *Actuators* **2020**, *9*, 113.
https://doi.org/10.3390/act9040113

**AMA Style**

Antonelli MG, Beomonte Zobel P, D’Ambrogio W, Durante F. Design Methodology for a Novel Bending Pneumatic Soft Actuator for Kinematically Mirroring the Shape of Objects. *Actuators*. 2020; 9(4):113.
https://doi.org/10.3390/act9040113

**Chicago/Turabian Style**

Antonelli, Michele Gabrio, Pierluigi Beomonte Zobel, Walter D’Ambrogio, and Francesco Durante. 2020. "Design Methodology for a Novel Bending Pneumatic Soft Actuator for Kinematically Mirroring the Shape of Objects" *Actuators* 9, no. 4: 113.
https://doi.org/10.3390/act9040113