# A Miniature Pneumatic Bending Rubber Actuator Controlled by Using the PSO-SVR-Based Motion Estimation Method with the Generalized Gaussian Kernel

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structure of a Miniature Pneumatic Bending Rubber Actuator

## 3. Mathematical Preliminaries

#### 3.1. Support Vector Regression

#### 3.2. Generalized Gaussian Distribution

#### 3.3. Particle Swarm Optimization

- Initialize the particle’s position and velocity in the swarm randomly, and set ranges of position and velocity.
- Evaluate positions with the evaluation function.
- Update the particle’s and global best solution.
- Update the particle’s velocity and position.

## 4. Modeling and Nonlinear Control System

#### 4.1. Modeling

#### 4.2. Design for the Operator-Based Robust Nonlinear Control System

#### 4.3. Tracking Actuator’s Output for the Target Value

## 5. Proposed Method

- Input training data for SVR with the generalized Gaussian kernel.
- Do many tests and evaluations with various parameters.
- Decide the best parameters.
- Estimate the actuator’s output by SVR with the optimized parameters.

## 6. Experiment

#### 6.1. Experimental System

- Compressed air is made by the air compressor.
- Air pressure is regulated by the regulator to prevent the actuator from breaking.
- Controlled air pressure is made to control the actuator by the electro-pneumatic regulator.
- Air pressure is provided for the actuator, and it moves.

#### 6.2. Experimental Result

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

SVM | Support vector machine |

SVR | Support vector regression |

PSO | Particle swarm optimization |

GD | Gaussian distribution |

GGD | Generalized Gaussian distribution |

FEM | Finite element method |

## References

- Wakimoto, S.; Suzumori, K.; Ogura, K. Miniature pneumatic curling rubber actuator generating bidirectional motion with one air-supply tube. Adv. Rob.
**2011**, 25, 1311–1330. [Google Scholar] [CrossRef] - Vapnik, N.V. Statistical Learning Theory; Springer: New York, NY, USA, 1998. [Google Scholar]
- Jiang, L.; Deng, M.; Inoue, A. Support vector machine-based two wheeled mobile robot motion control in noisy environment. J. Syst. Control Eng.
**2008**, 222, 733–743. [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] - Yu, S.; Zhang, A.; Li, H. A Review of Estimating the Shape Parameter of Generalized Gaussian Distribution. J. Comput. Inf. Syst.
**2012**, 8, 9055–9064. [Google Scholar] - Fujita, K.; Wakimoto, S.; Deng, M.; Wakitani, S. SVR-based input-output mapping of a micro-hand. In Proceedings of the 2015 International Conference on Advanced Mechatronic Systems, Beijing, China, 22–24 August 2015; pp. 538–541.
- Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia, 27 November–1 December 1995; pp. 1942–1948.
- Deng, M. Operator-Based Nonlinear Control Systems Design and Applications; Wiley-IEEE Press: Hoboken, 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] - Deng, M.; Bu, N. Robust Control for Nonlinear Systems Using Passivity-Based Robust Right Coprime Factorization. IEEE Trans. Autom. Control
**2012**, 57, 2599–2604. [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] - Iwai, Z.; Mizumoto, I.; Deng, M. Simple adaptive control of processes with time-delay. J. Process Control
**1997**, 7, 439–449. [Google Scholar]

**Figure 3.**PDFs of generalized Gaussian distribution (GGD) for $\alpha =0.7$, 1.0, 2.0, 100 $(\mu =0$, $\sigma =1)$.

Part of the Actuator | Parameter | Unit |
---|---|---|

Natural length of the actuator | ${L}_{0}$ | (m) |

Length of the changeless part | ${L}_{1}$ | (m) |

Length of the bellows side | L | (m) |

Radius of the approximate circle | R | (m) |

Bending angle | θ | (rad) |

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

Cost parameter | $C=100$ |

Error accuracy parameter | $\epsilon =0.001$ |

Variance | $\sigma =39.5712$ |

Shape parameter | $\alpha =2.027$ |

Paramter | Value |
---|---|

Natural length of the actuator | ${L}_{0}=1.35\times {10}^{-2}$ m |

Radius of the actuator | $a=1.0\times {10}^{-3}$ m |

Thickness of the actuator | $b=0.15\times {10}^{-3}$ m |

Rubber radius | ${r}_{a}=1.075\times {10}^{-3}$ m |

Initial Young modulus | ${E}_{0}=1.43\times {10}^{6}$ Pa |

Control parameter | $K=0.95$ |

Integral parameter | ${k}_{I}=8.0\times {10}^{-6}$ |

Proportional parameter | ${k}_{P}=5.0\times {10}^{-6}$ |

Proposed Method | Previous Method | |
---|---|---|

Number of dataset | 1080 | 5472 |

Computing time | $5.2$ s | $13.4$ s |

Fit ratio | $86.8\%$ | $84.8\%$ |

Proposed Method | Previous Method | |
---|---|---|

Number of dataset | 1437 | 5414 |

Computing time | $23.5$ s | $62.8$ s |

Fit ratio | $83.5\%$ | $-157\%$ |

© 2017 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**

Fujita, K.; Deng, M.; Wakimoto, S.
A Miniature Pneumatic Bending Rubber Actuator Controlled by Using the PSO-SVR-Based Motion Estimation Method with the Generalized Gaussian Kernel. *Actuators* **2017**, *6*, 6.
https://doi.org/10.3390/act6010006

**AMA Style**

Fujita K, Deng M, Wakimoto S.
A Miniature Pneumatic Bending Rubber Actuator Controlled by Using the PSO-SVR-Based Motion Estimation Method with the Generalized Gaussian Kernel. *Actuators*. 2017; 6(1):6.
https://doi.org/10.3390/act6010006

**Chicago/Turabian Style**

Fujita, Kou, Mingcong Deng, and Shuichi Wakimoto.
2017. "A Miniature Pneumatic Bending Rubber Actuator Controlled by Using the PSO-SVR-Based Motion Estimation Method with the Generalized Gaussian Kernel" *Actuators* 6, no. 1: 6.
https://doi.org/10.3390/act6010006