# System-Level Modelling and Simulation of a Multiphysical Kick and Catch Actuator System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Actuator System Design

- A sphere replaces the hemisphere;
- The sphere’s motion is restricted to vertical displacement;
- The kick-actuator comprises a single piezoelectric chip actuator;
- All of the components are arranged concentrically.

#### 2.2. Finite Element Models

#### 2.2.1. Electromagnetic Catch-Actuator

^{-6}or 30 iterations. To compute the electromagnetic force on the sphere with respect to its dependencies, both coil currents and the sphere’s position are parameterized. The sphere’s vertical position ranges from −2.5 $\mathrm{m}$$\mathrm{m}$ to 7.5 $\mathrm{m}$$\mathrm{m}$ in 41 steps, the coil currents from −0.1 $\mathrm{A}$ to 0.1 $\mathrm{A}$ in increments of $0.1$ $\mathrm{A}$. Finally, the catch-actuator is analyzed in a set of 369 static electromagnetic simulations. A more extensive description is available in [23,24].

#### 2.2.2. Piezoelectric Kick-Actuator

_{P}of ODEs expressed as:

#### 2.3. Model Order Reduction

#### 2.4. System-Level Simulation

## 3. Results

#### 3.1. Electromagnetic Force

#### 3.2. Reduced Order Model of the Kick-Actuator

#### 3.2.1. Harmonic Response

#### 3.2.2. Contact Force

#### 3.2.3. Transient Impact

#### 3.3. System-Level Simulation

## 4. Discussion and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DEIM | discrete empirical interpolation method |

DOF | degree of freedom |

EC | equivalent circuit |

ECSW | energy conserving mesh sampling and weighting |

FE | finite element |

FEM | finite element method |

FOM | full order model |

LiDAR | light detection and ranging |

MOR | model order reduction |

ODE | ordinary differential equation |

PDE | partial differential equation |

pMOR | parametric model order reduction |

POD | proper orthogonal decomposition |

PZT | lead zirconate titanate |

ROM | reduced order model |

## Appendix A. Material Data

#### Appendix A.1. Electromagnetic Material Properties

**Table A1.**Data for the linear materials used in the FE model of the catch actuator. ${\mathsf{\mu}}_{rel}$ is the relative permeability, ${B}_{r}$ the remanence, and ${H}_{c}$ the coercivity.

#### Appendix A.2. THP51

## Appendix B. Additional Harmonic Evaluations of the ROM

**Figure A1.**Harmonic relative error of each of the ROM’s outputs for a single input each. (

**a**) Unit force at ${R}_{1}$ (

**b**) Unit force at ${R}_{2}$ (

**c**) Unit force at ${R}_{3}$ (

**d**) Unit force at ${R}_{4}$ (

**e**) Unit force at ${R}_{5}$ (

**f**) Unit charge at the anode.

## References

- Chen, M.; Yu, H.; Guo, S.; Xu, R.; Shen, W. An electromagnetically-driven MEMS micromirror for laser projection. In Proceedings of the 10th IEEE International Conference on Nano/Micro Engineered and Molecular Systems, Xi’an, China, 7–11 April 2015; pp. 605–607. [Google Scholar] [CrossRef]
- Fan, K.C.; Lin, W.L.; Chiang, L.H.; Chen, S.H.; Chung, T.T.; Yang, Y.J. A 2 × 2 Mechanical Optical Switch With a Thin MEMS Mirror. J. Light. Technol.
**2009**, 27, 1155–1161. [Google Scholar] [CrossRef] - Hung, A.C.L.; Lai, H.Y.H.; Lin, T.W.; Fu, S.G.; Lu, M.S.C. An electrostatically driven 2D micro-scanning mirror with capacitive sensing for projection display. Sens. Actuators A Phys.
**2015**, 222, 122–129. [Google Scholar] [CrossRef] - Cho, A.R.; Han, A.; Ju, S.; Jeong, H.; Park, J.H.; Kim, I.; Bu, J.U.; Ji, C.H. Electromagnetic biaxial microscanner with mechanical amplification at resonance. Opt. Express
**2015**, 23, 16792–16802. [Google Scholar] [CrossRef] - Yalcinkaya, A.D.; Urey, H.; Brown, D.; Montague, T.; Sprague, R. Two-Axis Electromagnetic Microscanner for High Resolution Displays. J. Microelectromech. Syst.
**2006**, 15, 786–794. [Google Scholar] [CrossRef] - Seo, Y.H.; Hwang, K.; Kim, H.; Jeong, K.H. Scanning MEMS Mirror for High Definition and High Frame Rate Lissajous Patterns. Micromachines
**2019**, 10, 67. [Google Scholar] [CrossRef][Green Version] - Ju, S.; Jeong, H.; Park, J.H.; Bu, J.U.; Ji, C.H. Electromagnetic 2D Scanning Micromirror for High Definition Laser Projection Displays. IEEE Photonics Technol. Lett.
**2018**, 30, 2072–2075. [Google Scholar] [CrossRef] - Hwang, K.; Seo, Y.H.; Jeong, K.H. Microscanners for optical endomicroscopic applications. Micro Nano Syst. Lett.
**2017**, 5, 1. [Google Scholar] [CrossRef][Green Version] - Seo, Y.H.; Hwang, K.; Jeong, K.H. 1.65 mm diameter forward-viewing confocal endomicroscopic catheter using a flip-chip bonded electrothermal MEMS fiber scanner. Opt. Express
**2018**, 26, 4780–4785. [Google Scholar] [CrossRef] [PubMed] - Hu, Q.; Pedersen, C.; Rodrigo, P.J. Eye-safe diode laser Doppler lidar with a MEMS beam-scanner. Opt. Express
**2016**, 24, 1934–1942. [Google Scholar] [CrossRef][Green Version] - Kim, J.H.; Lee, S.W.; Jeong, H.S.; Lee, S.K.; Ji, C.H.; Park, J.H. Electromagnetically actuated 2-axis scanning micromirror with large aperture and tilting angle for lidar applications. In Proceedings of the 2015 Transducers—2015 18th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), Anchorage, AK, USA, 21–25 June 2015; pp. 839–842. [Google Scholar] [CrossRef]
- Royo, S.; Ballesta-Garcia, M. An Overview of Lidar Imaging Systems for Autonomous Vehicles. Appl. Sci.
**2019**, 9, 4093. [Google Scholar] [CrossRef][Green Version] - Wang, D.; Watkins, C.; Xie, H. MEMS Mirrors for LiDAR: A review. Micromachines
**2020**, 11, 456. [Google Scholar] [CrossRef] - Gu-Stoppel, S.; Giese, T.; Quenzer, H.J.; Hofmann, U.; Benecke, W. PZT-Actuated and -Sensed Resonant Micromirrors with Large Scan Angles Applying Mechanical Leverage Amplification for Biaxial Scanning. Micromachines
**2017**, 8, 215. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ou, C.H.; Lin, Y.C.; Keikoin, Y.; Ono, T.; Esashi, M.; Tsai, Y.C. Two-dimensional MEMS Fe-based metallic glass micromirror driven by an electromagnetic actuator. Jpn. J. Appl. Phys.
**2019**, 58, SDDL01. [Google Scholar] [CrossRef] - Park, Y.; Moon, S.; Lee, J.; Kim, K.; Lee, S.J.; Lee, J.H. Gimbal-Less Two-Axis Electromagnetic Microscanner with Twist Mechanism. Micromachines
**2018**, 9, 219. [Google Scholar] [CrossRef][Green Version] - Jia, K.; Pal, S.; Xie, H. An Electrothermal Tip–Tilt–Piston Micromirror Based on Folded Dual S-Shaped Bimorphs. J. Microelectromech. Syst.
**2009**, 18, 1004–1015. [Google Scholar] [CrossRef] - Lara-Castro, M.; Herrera-Amaya, A.; Escarola-Rosas, M.A.; Vázquez-Toledo, M.; López-Huerta, F.; Aguilera-Cortés, L.A.; Herrera-May, A.L. Design and Modeling of Polysilicon Electrothermal Actuators for a MEMS Mirror with Low Power Consumption. Micromachines
**2017**, 8, 203. [Google Scholar] [CrossRef][Green Version] - Markweg, E.; Nguyen, T.T.; Weinberger, S.; Ament, C.; Hoffmann, M. Development of a Miniaturized Multisensory Positioning Device for Laser Dicing Technology. Phys. Procedia
**2011**, 12, 387–395. [Google Scholar] [CrossRef][Green Version] - Bunge, F.; Leopold, S.; Bohm, S.; Hoffmann, M. Scanning micromirror for large, quasi-static 2D-deflections based on electrostatic driven rotation of a hemisphere. Sens. Actuators A Phys.
**2016**, 243, 159–166. [Google Scholar] [CrossRef] - DFG. Kick and Catch—Cooperative Microactuators for Freely Moving Platforms: SPP 2206: Cooperative Multilevel Multistable Micro Actuator Systems (KOMMMA), 2019. Available online: https://gepris.dfg.de/gepris/projekt/424616052 (accessed on 16 June 2021).
- Olbrich, M.; Schütz, A.; Kanjilal, K.; Bechtold, T.; Wallrabe, U.; Ament, C. Co-Design and Control of a Magnetic Microactuator for Freely Moving Platforms. Proceedings of 1st International Electronic Conference on Actuator Technology: Materials, Devices and Applications, Online, 23–27 November 2020; p. 8494. [Google Scholar] [CrossRef]
- Schütz, A.; Hu, S.; Rudnyi, E.B.; Bechtold, T. Electromagnetic System-Level Model of Novel Free Flight Microactuator. In Proceedings of the 2020 21st International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Microsystems (EuroSimE), Online, 6–27 July 2020; pp. 1–6. [Google Scholar] [CrossRef]
- Schütz, A.; Olbrich, M.; Hu, S.; Ament, C.; Bechtold, T. Parametric system-level models for position-control of novel electromagnetic free flight microactuator. Microelectron. Reliab.
**2021**, 119, 114062. [Google Scholar] [CrossRef] - Antoulas, A.C. Approximation of Large-Scale Dynamical Systems; Advances in design and control; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2005. [Google Scholar] [CrossRef]
- Rudnyi, E.B. MOR for ANSYS. In System-Level Modeling of MEMS; Bechtold, T., Schrag, G., Feng, L., Eds.; Advanced Micro and Nanosystems; Wiley-VCH-Verl.: Weinheim, Germany, 2013; pp. 425–438. [Google Scholar]
- Balajewicz, M.; Amsallem, D.; Farhat, C. Projection-based model reduction for contact problems. Int. J. Numer. Methods Eng.
**2016**, 106, 644–663. [Google Scholar] [CrossRef][Green Version] - Nasdala, L. (Ed.) Kontakt. In FEM-Formelsammlung Statik und Dynamik; Springer Fachmedien Wiesbaden: Wiesbaden, Germany, 2015; pp. 227–244. [Google Scholar] [CrossRef]
- Rutzmoser, J. Model Order Reduction for Nonlinear Structural Dynamics: Simulation-Free Approaches. Ph.D. Thesis, Technische Universität München, Garching, Germany, 2018. [Google Scholar]
- Carlberg, K.; Bou-Mosleh, C.; Farhat, C. Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. Int. J. Numer. Methods Eng.
**2011**, 86, 155–181. [Google Scholar] [CrossRef] - Goury, O.; Duriez, C. Fast, Generic, and Reliable Control and Simulation of Soft Robots Using Model Order Reduction. IEEE Trans. Robot.
**2018**, 34, 1565–1576. [Google Scholar] [CrossRef][Green Version] - Fauque, J.; Ramière, I.; Ryckelynck, D. Hybrid hyper-reduced modeling for contact mechanics problems. Int. J. Numer. Methods Eng.
**2018**, 115, 117–139. [Google Scholar] [CrossRef] - Tiso, P.; Rixen, D.J. Reduction methods for MEMS nonlinear dynamic analysis. In Nonlinear Modeling and Applications; Proulx, T., Ed.; Conference Proceedings of the Society for Experimental Mechanics Series; Springer: New York, NY, USA, 2011; Volume 2, pp. 53–65. [Google Scholar] [CrossRef]
- Idelsohn, S.R.; Cardona, A. A reduction method for nonlinear structural dynamic analysis. Comput. Methods Appl. Mech. Eng.
**1985**, 49, 253–279. [Google Scholar] [CrossRef] - Manvelyan, D.; Simeon, B.; Wever, U. An Efficient Model Order Reduction Scheme for Dynamic Contact in Linear Elasticity. arXiv
**2021**, arXiv:2102.03653. [Google Scholar] - Chaturantabut, S.; Sorensen, D.C. Nonlinear Model Reduction via Discrete Empirical Interpolation. SIAM J. Sci. Comput.
**2010**, 32, 2737–2764. [Google Scholar] [CrossRef] - Farhat, C.; Avery, P.; Chapman, T.; Cortial, J. Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency. Int. J. Numer. Methods Eng.
**2014**, 98, 625–662. [Google Scholar] [CrossRef] - Farhat, C.; Chapman, T.; Avery, P. Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models. Int. J. Numer. Methods Eng.
**2015**, 102, 1077–1110. [Google Scholar] [CrossRef] - Chapman, T. Nonlinear Model Order Reduction for Structural Systems with Contact. Ph.D. Thesis, Stanford University, Stanford, CA, USA, 2019. [Google Scholar]
- Del Tin, L. Reduced-Order Modelling, Circuit-Level Design and SOI Fabrication of Microelectromechanical Resonators. Ph.D. Thesis, Università di Bologna, Bologna, Italy, 2007. [CrossRef]
- Freund, R.W. Krylov-subspace methods for reduced-order modeling in circuit simulation. J. Comput. Appl. Math.
**2000**, 123, 395–421. [Google Scholar] [CrossRef][Green Version] - Thorlabs, Inc. Piezo Actuators, Brochure. Available online: https://www.thorlabs.com/images/Brochures/Thorlabs_Piezo_Brochure.pdf (accessed on 16 June 2021).
- Nasdala, L. (Ed.) FEM-Formelsammlung Statik und Dynamik; Springer Fachmedien Wiesbaden: Wiesbaden, Germany, 2015. [Google Scholar] [CrossRef]
- Bathe, K.J. Finite Element Procedures, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, USA, 2014. [Google Scholar]
- Thorlabs, Inc. PA3JEAW-SpecSheet: Piezoelectric Chip, 100 V, 2.2 μm Displacement, 3.0 ×3.0 × 2.0 mm, Pre-Attached Wires. Available online: https://www.thorlabs.com/thorproduct.cfm?partnumber=PA3JEAW (accessed on 16 June 2021).
- Bai, Z. Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math.
**2002**, 43, 9–44. [Google Scholar] [CrossRef][Green Version] - Hu, S.; Yuan, C.; Bechtold, T. Quasi-Schur Transformation for the Stable Compact Modeling of Piezoelectric Energy Harvester Devices. In Proceedings of the 12th International Conference on Scientific Computing in Electrical Engineering, Taormina, Italy, 23–27 September 2018; pp. 267–276. [Google Scholar] [CrossRef]
- Yuan, C.; Hu, S.; Castagnotto, A.; Lohmann, B.; Bechtold, T. Implicit Schur Complement for Model Order Reduction of Second Order Piezoelectric Energy Harvester Model. In Proceedings of the 9th Vienna International Conference on Mathematical Modelling (MATHMOD2018), Vienna, Austria, 21–23 February 2018. [Google Scholar] [CrossRef][Green Version]
- Hu, S.; Yuan, C.; Castagnotto, A.; Lohmann, B.; Bouhedma, S.; Hohlfeld, D.; Bechtold, T. Stable reduced order modeling of piezoelectric energy harvesting modules using implicit Schur complement. Microelectron. Reliab.
**2018**, 85, 148–155. [Google Scholar] [CrossRef] - Ansys
^{®}. Academic Research Electromagnetics Suite, Release 2020 R2, Canonsburg, PA, USA, 2020. Available online: https://www.ansys.com/products/release-highlights (accessed on 20 October 2021).

**Figure 1.**Working principle of the kick and catch actuator system: the kick-actuator launches the hemispherical micromirror into a flight phase. Subsequently, the electromagnetic catch-actuator controls the mirror’s flight. Finally, the catch-actuator decelerates the sphere and supports its smooth landing on the kick-actuator. This sequence achieves a small rotation of the hemisphere and may be repeated periodically to achieve large deflections. Please note the symbolic nature of this illustration. Later versions of the catch-actuator may, for example, contain a three-dimensional Helmholtz-coil configuration. Furthermore, this work focuses on further developing and applying the mathematical methodology of model order reduction (MOR). For this reason, the system is simplified to vertical motion.

**Figure 2.**Sectional three-dimensional view of the simplified actuator system with labeled components. This setup deploys a piezoelectric chip actuator for kick-actuation and an assembly of two coils and a ring magnet for electromagnetic interaction. Additionally, the micromirror is included as a magnetic sphere. The setup is designed for preliminary studies. This basis will be extended by more complex assemblies to precisely manipulate the micromirror.

**Figure 3.**Considerations for the FEM model of the piezoelectric kick-actuator: (

**a**) Symmetry allows to simulate only one quarter of the model, saving computational effort. (

**b**) A mapped mesh of concentric circles around the location of contact results in equal vertical nodal forces per ring. The center node and the five rings are enumerated from ${R}_{0}$ to ${R}_{5}$. (

**c**) Contact-induced forces on the kick-actuator are equal per ring. Black arrows indicate the vertical force distribution for a single ring. Separately modelling the sphere and the kick-actuator provides access to the MOR methodology proposed in [40].

**Figure 4.**Schematic workflow of MOR, illustrated for the Thorlabs PA3JEAW piezoelectric chip actuator: first, a physical problem to be investigated is chosen and subsequently modeled based on the FEM. The FEM spatially discretizes the computational domain, creating a large set of ODEs. MOR projects these ODEs onto a low-dimensional subspace, reducing the number of equations by several orders of magnitude. Finally, the resulting ROM is ready to use for commercial system-level simulation software. The picture of the actuator is adapted from [45] with the friendly permission of Thorlabs GmbH.

**Figure 5.**Schematic diagram of the kick and catch actuator system at system-level, extended by a PID-based position control of the sphere. The four grey areas indicate the system’s major components: the controller, the electromagnetic catch-actuator, the piezoelectric kick-actuator, and the sphere (clockwise from top left). The catch-actuator is modelled as an equivalent circuit, the kick-actuator as an ROM and the micromirror as a point mass.

**Figure 6.**The vertical electromagnetic force acting on the spherical magnet, plotted over its position. Each combination of coil currents is shown as a single line. The catch-actuator’s vertical symmetry induces a symmetric force for equal but opposed currents.

**Figure 7.**(

**a**) Comparison of the center node’s harmonic displacement amplitude obtained by the original FEM model and the ROM in a frequency range of 0 kHz to 500 kHz. (

**b**) The harmonic relative error of the ROM’s solution for all seven outputs, demonstrating its accuracy. The ROM approximates the original transfer function at an expansion point of 0 Hz. Consequently, all errors are the lowest at this frequency and increase with higher frequencies. Extending the reduced basis with vectors for higher frequencies enhances accuracy if needed. The error plots for the remaining six inputs are provided by Appendix B.

**Figure 8.**Static force opposing the sphere’s displacement into the kick-actuator. The sphere starts just in contact with the actuator’s top surface and is displaced 50 μm into the surface in increments of −0.2 μm. (

**a**) Solutions of the reference FEM model and the ROM. Note that the force is one quarter of the full contact force. (

**b**) The ROM’s relative error, demonstrating its accuracy.

**Figure 9.**The sphere’s vertical position during impact measured from its lowest point. (

**a**) The solutions of the reference FE model and the ROM. (

**b**) The ROM’s relative error increases in time as deviations accumulate. The sphere leaves contact after 11 $\mathsf{\mu}$$\mathrm{s}$, causing a high error at this point in time. Note that, due to limitations of commercial software, different methods are used for time integration.

**Figure 10.**The kick and catch actuation at system-level. (

**a**) The sphere’s vertical position over time for the full duration of 5 $\mathrm{m}$$\mathrm{s}$, showing a clear catch at $2.5$ $\mathrm{m}$$\mathrm{m}$. (

**b**) Vertical position of the center node for 450 $\mathsf{\mu}$$\mathrm{s}$–600 $\mathsf{\mu}$$\mathrm{s}$, illustrating the kick actuation.

**Table 1.**Dimensions of the catch actutator’s components. H corresponds to a part’s height, ${R}_{o}$ to the outer radius and ${R}_{i}$ to the inner radius. The vertical position y refers to center of mass.

Component | H [mm] | ${\mathit{R}}_{\mathit{o}}$ [mm] | ${\mathit{R}}_{\mathit{i}}$ [mm] | y [mm] |
---|---|---|---|---|

Ringmagnet | 1 | 5 | 4 | 2.5 |

Coil | 2 | 2.3 | 1.3 | 1/4 |

Sphere | - | 1 | - | - |

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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schütz, A.; Maeter, S.; Bechtold, T. System-Level Modelling and Simulation of a Multiphysical Kick and Catch Actuator System. *Actuators* **2021**, *10*, 279.
https://doi.org/10.3390/act10110279

**AMA Style**

Schütz A, Maeter S, Bechtold T. System-Level Modelling and Simulation of a Multiphysical Kick and Catch Actuator System. *Actuators*. 2021; 10(11):279.
https://doi.org/10.3390/act10110279

**Chicago/Turabian Style**

Schütz, Arwed, Sönke Maeter, and Tamara Bechtold. 2021. "System-Level Modelling and Simulation of a Multiphysical Kick and Catch Actuator System" *Actuators* 10, no. 11: 279.
https://doi.org/10.3390/act10110279