# Operator-Based Nonlinear Control for a Miniature Flexible Actuator Using the Funnel Control Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. The Structure of the Miniature Flexible Actuator

#### 2.2. Modeling of the Actuator Characteristics

#### 2.3. Modeling of the Pneumatic Characteristics

## 3. Nonlinear Control System Using the Funnel Control Method

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

#### 3.2. Passivity of the Proposed System

#### 3.3. Funnel Control

#### 3.4. Design Scheme of the Boundary Function

#### 3.4.1. Nonlinear Observer

#### 3.4.2. Boundary Function Using the Nonlinear Observers

## 4. Results and Discussion

#### 4.1. Experimental System

- The air compressor provides air pressure for the safety regulator.
- The air pressure is regulated by the safety regulator for the sake of not breaking the actuator.
- The computer sends an electrical signal to the electro-pneumatic regulator and decides on the opening of the valve.
- The air pressure is sent into the actuator and it moves.
- The output is captured as an image by a camera and fed back to the computer.

#### 4.2. Parameters Used in the Simulation and Experiment

#### 4.3. Simulation Results

#### 4.4. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FMA | Flexible micro-actuator |

## References

- Suzumori, K.; Iikura, S.; Tanaka, H. Development of flexible microactuator and its applications to robotic mechanisms. In Proceedings of the IEEE International Conference on Robotics and Automation, Sacramento, CA, USA, 9–11 April 1991; pp. 1622–1627. [Google Scholar] [CrossRef]
- Ichikawa, T.; Shintani, K.; Suzuki, T. Development of mechatronic esophagus using thin straight fibers type artifical muscle. Seisan Kenkyu
**2009**, 61, 135–138. [Google Scholar] - Noritsugu, T.; Tanaka, T. Application of rubber artificial muscle manipulator as a rehabilitation robot. IEEE/ASME Trans. Mechatronics
**1997**, 2, 259–267. [Google Scholar] [CrossRef] [Green Version] - Kawamura, S.; Sudani, M.; Deng, M.; Noge, Y.; Wakimoto, S. Modeling and system integration for a thin pneumatic rubber 3—DOF actuator. Actuators
**2019**, 8, 32. [Google Scholar] [CrossRef] [Green Version] - Tondu, B.; Lopez, P. Modeling and Control of McKibben artificial muscle Robot Actuators. IEEE Control Syst. Mag.
**2000**, 20, 15–38. [Google Scholar] - Itto, T.; Kogiso, K. Hybrid modeling of mckibben pneumatic artificial muscle systems. In Proceedings of the Joint IEEE International Conference on Industrial Technology Southeasetern Symposium on System Theory, Auburn, AL, USA, 14–16 March 2011; Volume 3, pp. 65–70. [Google Scholar]
- Nozaki, T.; Noritsugu, T. Motion analysis of McKibben type pneumatic rubber artificial muscle with finite element method. Int. J. Autom. Technol.
**2014**, 8, 147–158. [Google Scholar] [CrossRef] - Kawamura, S.; Deng, M. Recent Developments on Modeling for a 3—DOF Micro—Hand Based on AI Methods. Actuators
**2020**, 11, 792. [Google Scholar] [CrossRef] - Suzumori, K. Flexible Microactuator: 1st Report, static characteristics of 3 DOF actuator. Trans. Jpn. Soc. Mech. Eng. C
**1989**, 55, 2547–2552. (In Japanese) [Google Scholar] [CrossRef] [Green Version] - Wakimoto, S.; Suzumori, K.; Takeda, J. Flexible artificial muscle by bundle of McKibben fiber actuators. IEEE/ASME Int. Conf. Adv. Intell. Mechatronics
**2011**, 457–462. [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] [Green Version] - Wakimoto, S.; Suzumori, K.; Nishioka, Y. Miniaturization of large displacement rubber actuator. JSME Bioeng. Conf.
**2011**, 22, 104. [Google Scholar] [CrossRef] [Green Version] - Sudani, M.; Deng, M.; Wakimoto, S. Modeling and operator-based nonlinear control for a miniature pneumatic bending rubber actuator considering bellows. Actuators
**2018**, 7, 26. [Google Scholar] [CrossRef] [Green Version] - Deng, M.; Ueno, K. Operator-based nonlinear position control for a micro-hand by using image information. In Proceedings of the 2017 International Conference on Advanced Mechatronic Systems, Xiamen, China, 6–9 December 2017; pp. 46–50. [Google Scholar] [CrossRef] [Green Version]
- 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, 6. [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.
**2015**, 24, 318–324. [Google Scholar] [CrossRef] - Deng, M. Operator-Based Nonlinear 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] [Green Version] - Chen, G.; Han, Z. Robust right coprime factorization and robust stabilization of nonlinear feedback control system. IEEE Trans. Autom. Control
**1998**, 43, 1505–1509. [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] - Bu, N.; Deng, M. Passivity—Based Tracking Control for Uncertain Nonlinear Feedback Systems. J. Robot. Mechatronics
**2016**, 28, 837–841. [Google Scholar] [CrossRef] - Deng, M.; Ueno, K. Experimental Study on Operator–based Nonlinear Control for a Miniature Pneumatic Bending Rubber Actuator by Using PSO–SVR–GGD Method. In Proceedings of the 2019 IEEE 16th International Conference on Networking, Sensing and Control, Banff, AB, Canada, 9–11 May 2019; pp. 317–322. [Google Scholar] [CrossRef]
- Ilchmann, A.; Ryan, E.P.; Sangwin, C.J. Tracking with prescribed transient behaviour. ESAIM Control Optim. Calc. Var.
**2002**, 7, 471–493. [Google Scholar] [CrossRef] [Green Version] - Ilchmann, A.; Ryan, E.P.; Trenn, S. Tracking control: Performance funnels and prescribed transient behaviour. Syst. Control Lett.
**2005**, 7, 655–670. [Google Scholar] [CrossRef] [Green Version] - Efimov, D.; Raïssi, T.; Chebotarev, S.; Zolghadri, A. Interval state observer for nonlinear time varying systems. Automatica
**2013**, 49, 200–205. [Google Scholar] [CrossRef] [Green Version] - Shimizu, K. Nonlinear state observers by gradient descent method. In Proceedings of the IEEE International Conference on Control Applications, Anchorage, AK, USA, 27 September 2000; pp. 616–622. [Google Scholar] [CrossRef]

**Figure 12.**The error e and boundary functions $\overline{\mathcal{F}}$ and $\underline{\mathcal{F}}$ of the proposed method.

**Figure 13.**Enlarged view of Figure 12.

**Figure 22.**The error e and boundary functions $\overline{\mathcal{F}}$ and $\underline{\mathcal{F}}$ of the proposed method.

**Figure 23.**Enlarged view of Figure 22.

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

${L}_{0}$ | Initial length of the actuator | [m] |

${t}_{th}$ | Thickness of the rubber | [m] |

${r}_{1}$ | Internal radius of small chambers | [m] |

${R}_{1}$ | Representative radius of small chambers | [m] |

${r}_{2}$ | Internal radius of large chambers | [m] |

${R}_{2}$ | Representative radius of large chambers | [m] |

n | Number of the bellows | [-] |

E | Young’s modulus | [Pa] |

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

P | Air pressure of the actuator | [Pa] |

R | Gas constant | [J/Kg$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$K] |

T | Absolute temperature of air | [K] |

k | Heat capacity ratio of air | [-] |

V | Volume of the actuator | [${\mathrm{m}}^{3}$] |

m | Air flow rate | [Kg] |

${A}_{0}$ | Cross-sectional area of the control valve | [${\mathrm{m}}^{2}$] |

${P}_{tank}$ | Internal pressure of the compressor | [Pa] |

u | Input current | [mA] |

$\beta $ | Parameter of the control valve | [Pa/mA] |

$\gamma $ | Parameter of the control valve | [Pa] |

${P}_{max}$ | Maximum output pressure of the control valve | [Pa] |

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

${L}_{0}$ | Initial length of the actuator | $0.6\times {10}^{-3}$ m |

${t}_{th}$ | 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 | 12 |

E | Young’s modulus | $0.95\times {10}^{6}$ Pa |

$\alpha $ | Parameter of the control valve | $0.34$ |

$\beta $ | Parameter of the control valve | $6.25$ Pa/mA |

$\gamma $ | Parameter of the control valve | 25 Pa |

K | Control parameter | 40 |

${K}_{P}$ | Proportional parameter | $0.19$ |

${K}_{I}$ | Integral parameter | $0.19$ |

$\overline{\mathcal{L}}$ | Designed parameter | 1 |

$\underline{\mathcal{L}}$ | Designed parameter | 1 |

$\overline{\Delta}$ | Designed parameter | $0.05$ |

$\underline{\Delta}$ | Designed parameter | $0.1$ |

$\overline{\delta}$ | Designed parameter | $0.3$ |

$\underline{\delta}$ | Designed parameter | $-0.3$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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**

Ueno, K.; Kawamura, S.; Deng, M.
Operator-Based Nonlinear Control for a Miniature Flexible Actuator Using the Funnel Control Method. *Machines* **2021**, *9*, 26.
https://doi.org/10.3390/machines9020026

**AMA Style**

Ueno K, Kawamura S, Deng M.
Operator-Based Nonlinear Control for a Miniature Flexible Actuator Using the Funnel Control Method. *Machines*. 2021; 9(2):26.
https://doi.org/10.3390/machines9020026

**Chicago/Turabian Style**

Ueno, Keisuke, Shuhei Kawamura, and Mingcong Deng.
2021. "Operator-Based Nonlinear Control for a Miniature Flexible Actuator Using the Funnel Control Method" *Machines* 9, no. 2: 26.
https://doi.org/10.3390/machines9020026