# Characterization and Analysis of Extensile Fluidic Artificial Muscles

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. FAM Static Behavior

#### 1.2. Literature on FAM Modeling

#### 1.3. Content Overview

## 2. Fabrication

## 3. Experimental Characterization

#### 3.1. Experimental Setup

#### 3.2. Measurements

## 4. Modeling

#### 4.1. Resting Braid Angle Estimation from Geometric Properties

#### 4.2. Working Volume Estimation

#### 4.3. Dead-Band Pressure Estimation

#### 4.4. Resting Braid Angle Estimation from Experimental Data

^{®}’s fminsearch to find a braid angle that matches the blocked force data with Gaylord’s force model (Figure 8). The experimental values that do not follow the linear trend of Gaylord’s model are treated as outliers. Alternatively, an average value of the tangent of resting braid angle $tan\left({\theta}_{rest}\right)$ can also be obtained from data when one isolates $tan\left({\theta}_{rest}\right)$ from Gaylord’s model. It was found that the first experimental method was simpler to implement, and chose it because it directly minimizes the error, that depends on the braid angle, between Gaylord’s model and the experimental data on blocked forces. Blocked force value for FAM2 at a test pressure of $413.7$ kPa does not follow the trend and was treated as an outlier.

#### 4.5. Calculation of FAM Outer Radius and Bladder Thickness

#### 4.6. Numerical Estimation of Bladder Material Stress Function

#### 4.7. Stress Function Model Structures

#### 4.8. Modeling Results

## 5. Discussion

## 6. Conclusions

## 7. Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FAM | Fluidic artificial muscle |

CFAM | Contractile fluidic artificial muscle |

EFAM | Extensile fluidic artificial muscle |

PWM | Pulse width modulation |

PI | Proportional-intergral |

LTI | Linear time-invariant |

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**Figure 1.**(

**a**) Overview of a fluidic artificial muscle. Top: exploded view; bottom: assembly. A fluidic artificial muscle (FAM) is composed of three main components: braided sleeve, elastomeric bladder, and end fittings. End fittings fix the ends of the sleeve and bladder together, and seal the FAM on both ends. The braid angle of the sleeve is dependent on the FAM’s length and diameter. (

**b**) Half cross-sectional view of a FAM with swaged end fittings. The elastomeric bladder is surrounded by a braided sleeve. The bladder and the sleeve are fixed to the end fitting by “compressing” the bladder and the sleeve between a swaged tube and the core of the end fitting. Pressurized fluid flows in and out through a threaded channel in the end fitting. The end fitting can be attached to a load with a hollow and threaded stud fastened to the channel.

**Figure 2.**The figure shows contractile and extensile FAMs in the undeformed and deformed states. Here, FAMs are idealized as tubes of variable length and diameter with a geometric constraint imposed by an elliptical thread. Different braid angles of the thread result in two directions of motion. (

**a**) For braid angle ${\theta}_{c}$ > 35.26°, the inner ${R}_{ci}$ and outer ${R}_{c0}$ radii increase with pressure and the FAM length decreases together with the braid angle ${\theta}_{c}$. Due to the length decrease, the FAM is classified as contractile. (

**b**) For ${\theta}_{e}$ < 35.26°, the radii decrease with added pressure, and the FAM length increases along with the braid angle ${\theta}_{e}$ to form an extensile FAM.

**Figure 3.**The relationship between the absolute normalized blocked force of Gaylord’s model and the resting braid angle. A FAM blocked force is non-linearly dependent on its resting braid angle.

**Figure 4.**Manufactured and cycled extensile fluidic artificial muscle. The excess of the sleeve accumulated in arbitrary regions along the actuator. The diameter was not uniform along the axis of the specimen.

**Figure 5.**(

**a**) Test setup schematic. Dashed lines represent electric signals. Solid lines represent pneumatic connections. Pressurized air is supplied by the air compressor and then regulated to $827.4$ kPa (120 psi) by the pressure regulator. The proportional valve directs and tunes the air flow in or out of the FAM. FAM inflation pressure is read by the pressure transducer and split to be recorded by both the MTS control unit and the Arduino board. The Arduino board runs a proportional-integral (PI) controller that is fed the error between user-set pressure value and the FAM’s pressure. The PI controller sets the control signal for the proportional valve, which in turn increases or decreases the pressure with varying air flow. The EFAM generates compression on the load cell, which depends on both EFAM internal pressure and displacement (or stroke) of the MTS machine. Time, compressive force, pressure, and displacement are recorded by the MTS control unit to form a dataset for a given experiment. (

**b**) The extensile fluidic artificial muscle (EFAM) being tested.

**Figure 6.**Quasi-static axial force response plot under isobaric conditions of FAM1 and FAM2. The data show hysteresis between parts of testing cycles with increasing and decreasing length change.

**Figure 7.**For the two extensile FAMs, the blocked force trend is strongly linear, and increases with pressure. Free extension also increases with pressure to reach saturation above 517.1 kPa (75 psi). The blue and orange lines correspond to FAM1 and FAM2, respectively.

**Figure 8.**For the two extensile FAMs, blocked force was plotted based on the Gaylord’s force model, Equation (9). Braid angle was found based on geometric properties of fabricated FAMs, and with error minimization with respect to the experimental data.

**Figure 9.**The quasi-static axial force responses of FAM1 extensile fluidic muscles to varying strain for incrementally increasing internal pressure. The axial force outputs of the force-balance model employing the studied stress functions are overlayed as black lines for comparison to experimental data.

**Figure 10.**The quasi-static axial force responses of FAM2 extensile fluidic muscles to varying strain for incrementally increasing internal pressure. The axial force outputs of the force-balance model employing the studied stress functions are overlayed as black lines for comparison to experimental data.

Part Name | Quantity | Material | Length, cm (in) | Diameter, cm (in) |
---|---|---|---|---|

End fitting | 2 | Aluminium | 2.54 (1) | 2.133 (0.84) |

Swage tube | 2 | Aluminium | 2.54 (1) | inner: 2.25 (0.884), outer: 2.54 (1) |

Elastomeric tube | 1 | Soft latex rubber | 17.78 (7) | inner: 1.9 (3/4), outer: 2.22 (7/8) |

Braided sleeve | 1 | Kevlar (aramid) biaxial | 50.8 (20) at $\theta =45$° | 2.22 (7/8) at $\theta =45$° |

Binder/seal | N/A | Epoxy/hardener | N/A | N/A |

**Table 2.**The table shows constants that were measured during manufacturing or provided by components’ distributors. All the constants were later used for modeling.

Constant | Used To Calculate | Measure |
---|---|---|

${L}_{B}$—bladder length | Working bladder volume | Before assembly |

${D}_{{B}_{IN}}$—bladder inner diameter | Working bladder volume | Provided by manufacturer |

${D}_{{B}_{OUT}}$—bladder outer diameter | Working bladder volume | Provided by manufacturer |

${t}_{TH}$—thread thickness | Working bladder volume | Before assembly |

${L}_{45}$—braided sleeve length at $\theta =45$° | Braid angle, braid length | Before assembly |

${D}_{45}$—braided sleeve diameter at $\theta =45$° | Braid angle, blocked force | Provided by manufacturer |

${L}_{rest}$—resting length | Braid angle, blocked force | When assembled and cycled |

${D}_{rest}$—resting outer diameter | Braid angle, blocked force | When assembled and cycled |

**Table 3.**Error Comparison for the four different stress function models. Two error values are shown for FAM1 and FAM2, respectively.

Name: | Linear | Polynomial | Decaying Exponential | Exponent |
---|---|---|---|---|

Function Structure: | $({E}_{1}+{E}_{2}P)\u03f5$ | $({E}_{1}+{E}_{2}P)\u03f5$ | $({E}_{1}+{E}_{2}P)\u03f5$ | $({E}_{1}+{E}_{2}P)\u03f5$ |

$+{E}_{3}{\u03f5}^{2}+{E}_{4}{\u03f5}^{3}$ | $+{E}_{3}\left(1-{e}^{-{E}_{4}\u03f5}\right)$ | $+{E}_{1}({\u03f5}^{{E}_{3}}+{\u03f5}^{{E}_{4}})$ | ||

Comparison Length | Error, % | |||

Blocked Force * | 2.59, 4.02 | 2.59, 4.01 | 2.6, 3.99 | 2.79, 3.71 |

$\frac{2}{5}\Delta {L}_{max}$ | 8.75 **, 5.51 ** | 1.82, 2.37 | 1.83, 2.37 | 1.83, 2.37 |

$\frac{3}{5}\Delta {L}_{max}$ | 6.71, 4.59 | 3.09, 2.23 | 2.32, 1.51 | 2.26, 1.49 |

$\frac{4}{5}\Delta {L}_{max}$ | 2.32, 0.72 | 2.17, 0.73 | 2.16, 0.74 | 2.16, 0.74 |

$\Delta {L}_{max}$ | 4.46, 1.48 | 1.22, 0.61 | 1.22, 0.62 | 1.23, 0.62 |

Average | 4.97 **, 3.26 ** | 2.18, 1.99 | 2.03, 1.85 | 2.05, 1.79 |

FAM | Model | Function Structure | ${\mathit{E}}_{1}\xb7{10}^{6}$ | ${\mathit{E}}_{2}$ | ${\mathit{E}}_{3}$ | ${\mathit{E}}_{4}$ |
---|---|---|---|---|---|---|

1 | Linear | $({E}_{1}+{E}_{2}P)\u03f5$ | 1.766 | 1.5717 | ||

2 | Linear | $({E}_{1}+{E}_{2}P)\u03f5$ | 1.638 | 2.2113 | ||

1 | Polynomial | $({E}_{1}+{E}_{2}P)\u03f5+{E}_{3}{\u03f5}^{2}+{E}_{4}{\u03f5}^{3}$ | 2.367 | 1.9749 | −1,822,118 | 1,151,902 |

2 | Polynomial | $({E}_{1}+{E}_{2}P)\u03f5+{E}_{3}{\u03f5}^{2}+{E}_{4}{\u03f5}^{3}$ | 2.086 | 2.344 | −1,365,659 | 947,580 |

1 | Exponential-Decay | $({E}_{1}+{E}_{2}P)\u03f5+{E}_{3}\left(1-{e}^{-{E}_{4}\u03f5}\right)$ | 1.479 | 1.9771 | 121,646 | 13.0762 |

2 | Exponential-Decay | $({E}_{1}+{E}_{2}P)\u03f5+{E}_{3}\left(1-{e}^{-{E}_{4}\u03f5}\right)$ | 1.468 | 2.3456 | 74976 | 28.7297 |

1 | Exponent | $({E}_{1}+{E}_{2}P)\u03f5+{E}_{1}({\u03f5}^{{E}_{3}}+{\u03f5}^{{E}_{4}})$ | 0.538 | 1.9782 | 0.5061 | 1.3539 |

2 | Exponent | $({E}_{1}+{E}_{2}P)\u03f5+{E}_{1}({\u03f5}^{{E}_{3}}+{\u03f5}^{{E}_{4}})$ | 0.531 | 2.346 | 0.5599 | 1.5233 |

**Table 5.**Comparison of constitutive models. Models were compared with: their average error, if there was a simple expression for each coefficient, or a multi-coefficient expression was needed to satisfy the material stability criterion, growth rate in Big O notation, and for which coefficients boundaries are known. Bndd stands for bounded, and lbnd stands for lower bound.

Model | Average Error, % | Stability Conditions | $\mathcal{O}\left(\mathit{\sigma}\right(\mathit{\u03f5}\left)\right)$ | Known Coef. Boundaries |
---|---|---|---|---|

Linear | 4.12 | One-coeff. inequalities | $\mathcal{O}\left(\u03f5\right)$ | ${E}_{1}$ (bndd); ${E}_{2}$ (lbnd) |

Polynomial | 2.09 | Multi-coeff. inequality | $\mathcal{O}\left({\u03f5}^{3}\right)$ | ${E}_{2}$ (lbnd) |

Exponential-Decay | 1.94 | One-coeff. inequalities | $\mathcal{O}\left(\u03f5\right)$ | ${E}_{1}$, ${E}_{3}$ (bndd); ${E}_{2}$, ${E}_{4}$ (lbnd) |

Exponent | 1.92 | One-coeff. inequalities | $\mathcal{O}\left({\u03f5}^{{E}_{4}}\right)$ | ${E}_{3}$ (bndd); ${E}_{2}$, ${E}_{4}$ (lbnd) |

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**MDPI and ACS Style**

Garbulinski, J.; Balasankula, S.C.; Wereley, N.M.
Characterization and Analysis of Extensile Fluidic Artificial Muscles. *Actuators* **2021**, *10*, 26.
https://doi.org/10.3390/act10020026

**AMA Style**

Garbulinski J, Balasankula SC, Wereley NM.
Characterization and Analysis of Extensile Fluidic Artificial Muscles. *Actuators*. 2021; 10(2):26.
https://doi.org/10.3390/act10020026

**Chicago/Turabian Style**

Garbulinski, Jacek, Sai C. Balasankula, and Norman M. Wereley.
2021. "Characterization and Analysis of Extensile Fluidic Artificial Muscles" *Actuators* 10, no. 2: 26.
https://doi.org/10.3390/act10020026