# Efficient Structure-Based Models for the McKibben Contraction Pneumatic Muscle Actuator: The Full Description of the Behaviour of the Contraction PMA

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structure of the Pneumatic Muscle Actuator

_{0}, D

_{0}, and θ

_{0}, respectively.

## 3. Operation of the PMA

_{0}is the initial actuator length and L is the length of the PMA under the pressurised condition.

_{1}, a contracting force will start to lift the fixed load until it reaches the balance point; at this point the contraction force is similar to the mass weight [16].

_{1}and the length reduces to L

_{1}. An increment in the pressure to P

_{2}leads to an increase in the actuator’s volume and creates more contraction to L

_{2}, and the air pressure will increase to its maximum value which is subject to the structure of the actuator.

_{1}to M

_{2}, and then to no load, will raise the volume and reduce the length of the actuator [16]. The multiple lengths and volumes produced depend on the amount of the air pressure inside the muscle.

## 4. Presented Force Formula

_{1}and c

_{2}are positive constants. From this equation, the correction factor becomes 1 at maximum air pressure, where the actuator shape is cylindrical.

_{in}) for the McKibben’s muscle under air pressure supply is:

_{out}) occurs when the actuator shortens with the volume change:

^{3}, D

_{in}is the inner diameter in m, L is the length of the PMA in m, D

_{out}is the outer diameter in m, and Th

_{D}is twice the value of both the inner rubber tube and the braided sleeve thickness. Figure 5 shows the cross-section of the actuator structure.

_{in}will be less and depends on the thickness of the rubber tube and the braided sleeve. Increasing the rubber tube stiffness leads to increase its resistance and the W

_{out}will decrease, while the generated pulling force affects it longitudinally.

_{rs}is the resistance force (N) of the rubber tube, s

_{r}is the stiffness (N/m) of the rubber tube, A

_{in}(m

^{2}) is the inner area of the rubber cross-section, and ΔL (m) is the change of the actuator length between the initial length and length at each pressure step.

- An air pressure is applied to the actuator in Table 1 with step values from 0 to 5 bar.
- At each step, the input work is calculated from Equations (3) and (8).
- Subtracting the losses due to the stiffness and the changing in the inner area using Equation (10).
- Calculating the output work by multiply the pulling force of the actuator by the length change of Equation (4) (see Figure 6).
- Repeat this experiment on different actuators.
- Fixed value of work losses are found to be about 0.641 Nm and it is considered to be due to the contactless losses between the inner tube and the braided sleeve, and the force losses decrease when the ($\mathsf{\u2206}L$) increases (i.e., the contact occurs when the pressure is increased).

_{g}< 0.5 bar) and its value has been decreased dramatically at P

_{g}= 0.5 bar.

_{g}< 0.5 bar is significantly high.

_{rs}and f

_{c}is high at P

_{g}< 0.5 bar, the pulling force F cannot be produced and the generated force increases when the opposing forces are decreased.

_{c}) for this muscle have the same values as the pressure seen in Figure 8, while the resistance force (f

_{rs}) is different because it is dependent on the rubber’s stiffness. Figure 10 shows the resistance force against the air pressure. The force of this actuator is higher than the force of the first PMA while the losses are increased because the stiffness is increased.

_{rs}can be given as in Figure 11. The mean square error (MSE) is listed in Table 4 and gives the error between the experimental and the proposed force formula to the actuators understudy for the pressure steps from 0 to 0.5 bar and it is calculated according to Equation (13):

## 5. Structure-Based Length Formula for Contraction PMAs

_{0}and they represent the coefficients of Equation (14).

_{0}), initial diameter (D

_{0}), and the rubber stiffness must be considered. While the contraction ratio increases as a pressure increase, and the highest rubber stiffness s

_{r}leads to the lowest stretchable ability, then:

## 6. The Stiffness of the Contraction PMA

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 8.**The contactless losses between the rubber tube and the braided sleeve against air pressure.

**Figure 12.**The experimental and theoretical length of the PMAs. (

**A**–

**D**) represent the length for the PMAs in Table 5 respectively.

**Figure 14.**The experimental and theoretical stiffness of the three different PMAs. (

**A**–

**C**) represent the stiffness for the PMAs in Table 5 respectively.

L_{0} (m) | Rubber Thickness (m) | Braided Thickness (m) | Inner Diameter (m) | Rubber Stiffness (N/m) |
---|---|---|---|---|

0.2 | 1.1 × 10^{−3} | 0.5 × 10^{−3} | 12 × 10^{−3} | 363.33 |

L_{0} (m) | Rubber Thickness (m) | Braided Thickness (m) | Inner Diameter (m) | Rubber Stiffness (N/m) |
---|---|---|---|---|

0.2 | 2.2 × 10^{−3} | 0.5 × 10^{−3} | 12 × 10^{−3} | 1090 |

L_{0} (m) | Rubber Thickness (m) | Braided Thickness (m) | Inner Diameter (m) | Rubber Stiffness (N/m) |
---|---|---|---|---|

0.2 | 1.1 × 10^{−3} | 0.5 × 10^{−3} | 26.5 × 10^{−3} | 545 |

Actuator | MSE |
---|---|

1 | 7.23 |

2 | 9.31 |

3 | 22.68 |

PMA | L_{0} (m) | Actuator Diameter (m) | Rubber Stiffness (N/m) |
---|---|---|---|

A | 0.2 | 0.0152 | 363.33 |

B | 0.2 | 00.0174 | 1090 |

C | 0.2 | 0.0297 | 545 |

D | 0.3 | 0.0152 | 363.33 |

PMA | Contraction Ratio (ε) |
---|---|

A | 29% |

B | 19.5% |

C | 24.3% |

D | 28.6% |

Load (kg) | MSE |
---|---|

0.0 | 0.01982 |

0.5 | 0.1405 |

1.0 | 0.1995 |

2.0 | 0.2363 |

3.0 | 0.431 |

4.0 | 0.9697 |

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**MDPI and ACS Style**

Al-Ibadi, A.; Nefti-Meziani, S.; Davis, S. Efficient Structure-Based Models for the McKibben Contraction Pneumatic Muscle Actuator: The Full Description of the Behaviour of the Contraction PMA. *Actuators* **2017**, *6*, 32.
https://doi.org/10.3390/act6040032

**AMA Style**

Al-Ibadi A, Nefti-Meziani S, Davis S. Efficient Structure-Based Models for the McKibben Contraction Pneumatic Muscle Actuator: The Full Description of the Behaviour of the Contraction PMA. *Actuators*. 2017; 6(4):32.
https://doi.org/10.3390/act6040032

**Chicago/Turabian Style**

Al-Ibadi, Alaa, Samia Nefti-Meziani, and Steve Davis. 2017. "Efficient Structure-Based Models for the McKibben Contraction Pneumatic Muscle Actuator: The Full Description of the Behaviour of the Contraction PMA" *Actuators* 6, no. 4: 32.
https://doi.org/10.3390/act6040032