# Modelling and Operator-Based Nonlinear Control for a Miniature Pneumatic Bending Rubber Actuator Considering Bellows

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Structure of the Miniature Pneumatic Bending Rubber Actuator

#### 2.2. Modelling

#### 2.2.1. Neo-Hookean Model

#### 2.2.2. Relation between Input Pressure and Output Angle

#### 2.3. Operator-Based Nonlinear Control Feedback System Design

#### 2.3.1. Right Coprime Factorization

#### 2.3.2. Robust Stability

#### 2.3.3. Designing Operators

## 3. Results

#### 3.1. Experiment

- Convert captured color image to a gray scale.
- Dissolve the image into three pixel numbers; R, B, G.
- Extract only R pixels from the image in comparison with a gray scale.
- Find the center coordinate from extracted R pixel area.
- Do the same process as 4 and 5 to B pixels.
- Calculate bending angle from center coordinate of R and B area.

#### 3.2. Experimental Result

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations

FMA | Flexible micro actuator |

## References

- Chen, X.; Su, C.Y. Adaptive Control for Ionic Polymer-Metal Composite Actuators. In Proceedings of the 2016 IEEE Transactions on Systems, Man, and Cybernetics: Systems, Sacramento, CA, USA, 1 October 2016; Volume 46, pp. 1468–1477. [Google Scholar] [CrossRef]
- Suzumori, K.; Iikura, S.; Tanaka, H. Development of flexible microactutor and its applications to robotic mechanisms. In Proceedings of the IEEE International Conference on Robotics and Automation, Sacramento, CA, USA, 9–11 April 1991; Volume 4, pp. 1622–1627. [Google Scholar] [CrossRef]
- Wakimoto, S.; Suzumori, K.; Ogura, K. Miniature pneumatic curling rubber actuator generating bidirectional motion with one air-supply tube. Adv. Robot.
**2011**, 25, 1311–1330. [Google Scholar] [CrossRef] - Mosadegh, B.; Polygerinos, P.; Keplinger, C.; Wennstedt, S.; Shepherd, R.F.; Gupta, U.; Shim, J.; Bertoldi, K.; Walsh, C.J.; Whitesides, G.M. Pneumatic Networks for Soft Robotics that Actuate Rapidly. Adv. Funct. Mater
**2014**, 24, 2163–2170. [Google Scholar] [CrossRef] [Green Version] - Yoshioka, R.; Wakimoto, S.; Yamamoto, Y.; Suzumori, K. Development of a micro pneumatic actuator realizing bidirectional bending motions. In Proceedings of the 2013 JSME Conference on Robotics and Mechatronics, Tsukuba, Japan, 22–25 May 2013; Volume 5, pp. 2A1-D06(1)–2A1-D06(3). [Google Scholar] [CrossRef]
- Deng, M. Operator-Based Bonlinear Control Systems Design and Applications; Willy-IEEE Press: Piscataway, NJ, USA, 2014. [Google Scholar]
- Deng, M.; Inoue, A.; Ishikawa, K. Operator-based nonlinear feedback control design using robust right coprime factorization. IEEE Trans. Autom. Control
**2006**, 51, 645–648. [Google Scholar] [CrossRef] - Chen, G.; Han, Z. Robust right coprime factorization and robust stabilization of nonlinear feedback control system. IEEE Trans. Autom. Control
**1998**, 43, 645–648. [Google Scholar] [CrossRef] - Deng, M.; Bu, N.; Inoue, A. Output tracking of nonlinear feedback systems with perturbation based on robust right coprime factorization. Int. J. Innov. Comput. Inf. Control
**2009**, 5, 3359–3366. [Google Scholar] - Wang, A.; Deng, M. Robust nonlinear multivariable tracking control design to a manipulator with unknown uncertainties using operator-based robust right coprime factorization. Trans. Inst. Meas. Control
**2013**, 35, 788–797. [Google Scholar] [CrossRef] - Deng, M.; Bu, N. Robust control for nonlinear systems with unknown perturbations using simplified robust right coprime factorization. Int. J. Control
**2012**, 85, 842–850. [Google Scholar] [CrossRef] - Deng, M.; Kawashima, T. Adaptive nonlinear sensorless control for an uncertain miniature pneumatic curling rubber actuator using passivity and robust right coprime factorization. IEEE Trans. Control Syst. Technol.
**2016**, 24, 318–324. [Google Scholar] [CrossRef] - Miyagawa, T.; Toya, K.; Kubota, Y. Static characteristics of pneumatic soft actuator using fiber reinforced rubber. In Proceedings of the JSME Conference on Robotics and Mechatronics, Akita, Japan, 10–12 May 2007; pp. 1A2–A01(1)–1A2–A01(4). [Google Scholar] [CrossRef]
- Kim, B.; Lee, S.B.; Lee, J.; Cho, S.; Park, H.; Yeom, S.; Park, S.H. A comparison among neo-hookean model, mooney-rivlin model, and ogden model for chloroprene rubber. Int. J. Precis. Eng. Manuf.
**2012**, 13, 759–764. [Google Scholar] [CrossRef] - Editorial Board. Nyumonkouza yasashigomunobutsururi daisankou gomudanseitonendanseinokiso. [Introductory course of simple physics of rubber: No. 3 Basics of rubber elasticity and viscoelasticity]. J. SRI. Jpn.
**2007**, 80, 404–407. (In Japanese) [Google Scholar] - Fujita, K.; Deng, M.; Wakimoto, S. A Miniature bending rubber controlled by using the PSO-SVR-based motion estimation method with the generalized gaussian kernel. Actuators
**2017**, 6. [Google Scholar] [CrossRef]

**Figure 3.**(

**a**) The model for analysis; (

**b**) Three forces working on bellow; (

**c**) The strain on the outside compared with the inside; (

**d**) Three moments working on joint between small channel and large chamber.

**Figure 4.**(

**a**) ${f}_{1}$ working on a chamber or ${f}_{2}$ working on a channel; the chamber and the channel are geometry similar, so that ${f}_{1}$ and ${f}_{2}$ also work similarly; (

**b**) ${f}_{p}$ working on the surface.

**Figure 7.**The stabilized feedback system in the overall system of Figure 6.

Parameter | Definition | Value |
---|---|---|

l | Initial length of the actuator | $0.66\times {10}^{-3}$ m |

t | Thickness of the rubber | $0.15\times {10}^{-3}$ m |

${r}_{1}$ | Internal radius of small chambers | $0.25\times {10}^{-3}$ m |

${R}_{1}$ | Representative radius of small chambers | $0.325\times {10}^{-3}$ m |

${r}_{2}$ | Internal radius of large chambers | $0.85\times {10}^{-3}$ m |

${R}_{2}$ | Representative radius of large chambers | $0.925\times {10}^{-3}$ m |

n | Number of the bellows | 11 |

E | Young’s modulus | $0.96\times {10}^{6}$ Pa |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sudani, M.; Deng, M.; Wakimoto, S.
Modelling and Operator-Based Nonlinear Control for a Miniature Pneumatic Bending Rubber Actuator Considering Bellows. *Actuators* **2018**, *7*, 26.
https://doi.org/10.3390/act7020026

**AMA Style**

Sudani M, Deng M, Wakimoto S.
Modelling and Operator-Based Nonlinear Control for a Miniature Pneumatic Bending Rubber Actuator Considering Bellows. *Actuators*. 2018; 7(2):26.
https://doi.org/10.3390/act7020026

**Chicago/Turabian Style**

Sudani, Mizuki, Mingcong Deng, and Shuichi Wakimoto.
2018. "Modelling and Operator-Based Nonlinear Control for a Miniature Pneumatic Bending Rubber Actuator Considering Bellows" *Actuators* 7, no. 2: 26.
https://doi.org/10.3390/act7020026