# Robust Control of a Bimorph Piezoelectric Robotic Manipulator Considering Ellipsoidal-Type State Restrictions

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## Abstract

**:**

## 1. Introduction

## 2. Design of the Micromanipulation Device Using Piezoelectric Actuators

#### 2.1. Design of the Piezolectric Actuator-Based Robotic Manipulator

**Assumption**

**1.**

**Assumption**

**2.**

#### 2.2. Gripper Based on Parallel Arrange of Piezoelectric Actuators

#### 2.3. State Constraints for the Robotic Arm and the Microgripping Device

## 3. Problem Statement

## 4. Main Contribution of the Control Design

#### The Control Challenge

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

## 5. Experimental Validations of the Proposed Controller

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of Open Access Journals |

TLA | Three-Letter Acronym |

LD | Linear Dichroism |

## References

- Near, C.D. Piezoelectric actuator technology. Smart Structures and Materials 1996: Smart Structures and Integrated Systems. SPIE
**1996**, 2717, 246–258. [Google Scholar] - Tzen, J.J.; Jeng, S.L.; Chieng, W.H. Modeling of piezoelectric actuator for compensation and controller design. Precis. Eng.
**2003**, 27, 70–86. [Google Scholar] [CrossRef] - Richter, H.; Misawa, E.A.; Lucca, D.; Lu, H. Modeling nonlinear behavior in a piezoelectric actuator. Precis. Eng.
**2001**, 25, 128–137. [Google Scholar] [CrossRef] [Green Version] - Lining, S.; Changhai, R.; Weibin, R.; Liguo, C.; Minxiu, K. Tracking control of piezoelectric actuator based on a new mathematical model. J. Micromechan. Microeng.
**2004**, 14, 1439. [Google Scholar] [CrossRef] - Ru, C.; Chen, L.; Shao, B.; Rong, W.; Sun, L. A hysteresis compensation method of piezoelectric actuator: Model, identification and control. Control. Eng. Pract.
**2009**, 17, 1107–1114. [Google Scholar] [CrossRef] - Yi, J.; Chang, S.; Shen, Y. Disturbance-observer-based hysteresis compensation for piezoelectric actuators. IEEE/ASME Trans. Mechatronics
**2009**, 14, 456–464. [Google Scholar] - Zheng, Z.; Kumar, P.; Chen, Y.; Cheng, H.; Wagner, S.; Chen, M.; Verma, N.; Sturm, J.C. Piezoelectric Soft Robot Inchworm Motion by Controlling Ground Friction through Robot Shape. arXiv
**2021**, arXiv:2111.00944. [Google Scholar] - DeVoe, D.L.; Pisano, A.P. Modeling and optimal design of piezoelectric cantilever microactuators. J. Microelectromechanical Syst.
**1997**, 6, 266–270. [Google Scholar] [CrossRef] - Čeponis, A.; Jūrėnas, V.; Mažeika, D. Development of 5-DOF piezoelectric actuator for planar—Angular positioning. Appl. Sci.
**2022**, 12, 1033. [Google Scholar] [CrossRef] - Delibas, B.; Koc, B.; Thielager, J.; Stiebel, C. A novel drive and control method for piezoelectric motors in microscopy stages. In Proceedings of the Euspen’s 21st International Conference & Exhibition, Copenhagen, Denmark, 12–16 June 2021; pp. 7–10. [Google Scholar]
- Gao, X.; Yang, J.; Wu, J.; Xin, X.; Li, Z.; Yuan, X.; Shen, X.; Dong, S. Piezoelectric actuators and motors: Materials, designs, and applications. Adv. Mater. Technol.
**2020**, 5, 1900716. [Google Scholar] [CrossRef] - Touairi, S.; Khouya, Y.; Bahanni, C.; Khaouch, Z.; Mabrouki, M. Mechatronic control and modeling of a piezoelectric actuator. In Proceedings of the 2019 International Conference on Wireless Technologies, Embedded and Intelligent Systems (WITS), Fez, Morocco, 3–4 April 2019; pp. 1–6. [Google Scholar]
- Liu, R.; Wang, L.; Jin, J.; Zhao, H.; Zhang, A.; Chen, D. A novel 3-DoF piezoelectric robotic pectoral fin: Design, simulation, and experimental investigation. Smart Mater. Struct.
**2022**, 31, 65003. [Google Scholar] [CrossRef] - Yang, X.; Zhu, W.L.; Zhu, Z.; Zhu, L.M. Design, assessment, and trajectory control of a novel decoupled robotic nanomanipulator. IEEE/ASME Trans. Mechatronics
**2022**. [Google Scholar] [CrossRef] - Flores, G.; Aldana, N.; Rakotondrabe, M. Model predictive control based on the generalized Bouc-Wen model for piezoelectric actuators in robotic hand with only position measurements. IEEE Control. Syst. Lett.
**2021**, 6, 2186–2191. [Google Scholar] [CrossRef] - AbuZaiter, A.; Nafea, M.; Ali, M.S.M. Development of a shape-memory-alloy micromanipulator based on integrated bimorph microactuators. Mechatronics
**2016**, 38, 16–28. [Google Scholar] [CrossRef] - Flores, G.; Rakotondrabe, M. Robust nonlinear control for a piezoelectric actuator in a robotic hand using only position measurements. IEEE Control. Syst. Lett.
**2021**, 6, 872–877. [Google Scholar] [CrossRef] - Zhang, Z.; Wang, X.; Liu, J.; Dai, C.; Sun, Y. Robotic micromanipulation: Fundamentals and applications. Annu. Rev. Control. Robot. Auton. Syst.
**2019**, 2, 181–203. [Google Scholar] [CrossRef] - Yuan, S.; Zhu, C.; Chu, X.; Zhao, Y.; Amin, M.; Fan, Y. A novel linear piezoelectric actuator with two working principles of standing and traveling wave vibration mode. AIP Adv.
**2015**, 5, 107213. [Google Scholar] [CrossRef] [Green Version] - Liu, J.; Liu, Y.; Zhao, L.; Xu, D.; Chen, W.; Deng, J. Design and experiments of a single-foot linear piezoelectric actuator operated in a stepping mode. IEEE Trans. Ind. Electron.
**2018**, 65, 8063–8071. [Google Scholar] [CrossRef] - Liu, Z.; Yao, Z.; Jian, Y.; Zhang, B. Characteristics analysis of a plate type linear piezoelectric actuator based on a point contact model. Smart Mater. Struct.
**2018**, 27, 115031. [Google Scholar] [CrossRef] - Moghaddam, S.M.F.; Ahmadi, H. Active vibration control of truncated conical shell under harmonic excitation using piezoelectric actuator. Thin-Walled Struct.
**2020**, 151, 106642. [Google Scholar] [CrossRef] - Shirazi, M.J.; Salarieh, H.; Alasty, A.; Shabani, R. Tip tracking control of a micro-cantilever Timoshenko beam via piezoelectric actuator. J. Vib. Control.
**2013**, 19, 1561–1574. [Google Scholar] [CrossRef] - Bahrami, M.N.; Yousefi-Koma, A.; Raeisifard, H. Modeling and nonlinear analysis of a micro-switch under electrostatic and piezoelectric excitations with curvature and piezoelectric nonlinearities. J. Mech. Sci. Technol.
**2014**, 28, 263–272. [Google Scholar] [CrossRef] - Legnani, W.; Moschandreou, T.E.; Reyhanoglu, M. Nonlinear Systems: Theoretical Aspects and Recent Applications; IntechOpen: London, UK, 2020. [Google Scholar]

**Figure 1.**Description of the robotic arm and the gripping device based on bimorph piezoelectric ctuators.

**Figure 4.**(

**a**) Photograph of the electromechanical configuration for the first articulation ${\theta}_{1}$; (

**b**) comparison of the controlled motion for the first articulation (${\theta}_{1}$) when the PID and adaptive control with state constraints are considered.

**Figure 5.**(

**a**) Photograph of the electromechanical configuration for the second articulation ${\theta}_{2}$; (

**b**) comparison of the controlled motion for the second articulation (${\theta}_{2}$) when the PID and adaptive control with state constraints are considered.

**Figure 6.**(

**a**) Photograph of the electromechanical configuration for the third articulation ${\theta}_{3}$; (

**b**) comparison of the controlled motion for the third articulation (${\theta}_{3}$) when the PID and adaptive control with state constraints are considered.

**Figure 7.**Comparison of the mean square error between the proposed state constraint and the traditional state feedback control form.

**Figure 8.**Comparison of control action between the proposed state constraint and the traditional state feedback control form.

**Figure 9.**(

**a**) Direct visualization of the gripper section attached to the piezolectric actuators; (

**b**) manipulation of a 160 $\mathsf{\mu}$m polystyrene sphere using the gripping device made with the piezoelectric actuators.

Joint / Gain | ${\mathit{K}}_{\mathit{P}}$ | ${\mathit{K}}_{\mathit{D}}$ | ${\mathit{K}}_{\mathit{I}}$ |
---|---|---|---|

First | 90.92 | 5.21 | 3.4 |

Second | 80.92 | 15.21 | 4.6 |

Third | 72.34 | 24.12 | 7.6 |

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**MDPI and ACS Style**

Moreno-Guzman, F.; Salgado, I.; Cruz-Ortiz, D.; Chairez, I.
Robust Control of a Bimorph Piezoelectric Robotic Manipulator Considering Ellipsoidal-Type State Restrictions. *Appl. Sci.* **2022**, *12*, 7589.
https://doi.org/10.3390/app12157589

**AMA Style**

Moreno-Guzman F, Salgado I, Cruz-Ortiz D, Chairez I.
Robust Control of a Bimorph Piezoelectric Robotic Manipulator Considering Ellipsoidal-Type State Restrictions. *Applied Sciences*. 2022; 12(15):7589.
https://doi.org/10.3390/app12157589

**Chicago/Turabian Style**

Moreno-Guzman, Francisco, Ivan Salgado, David Cruz-Ortiz, and Isaac Chairez.
2022. "Robust Control of a Bimorph Piezoelectric Robotic Manipulator Considering Ellipsoidal-Type State Restrictions" *Applied Sciences* 12, no. 15: 7589.
https://doi.org/10.3390/app12157589