#
Co-Design and Control of a Magnetic Microactuator for Freely Moving Platforms^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Modelling

^{®}Maxwell [18] for a grid of system states and parameters and are implemented as a spline-interpolated lookup table. It is noteworthy that we can inversely compute the shape functions from (5) and (), given the numerical simulation data of the forces with known remanence values and currents. In the case that the permanent and electromagnet centre points are given by ${z}_{\mathrm{pm},j}$ and ${z}_{\mathrm{em}}$, the force can easily be obtained by shifting the shape function.

#### 2.2. Trajectory Planning and Control

#### 2.2.1. Electromagnetic Actuation

#### 2.2.2. Piezoelectric Actuation

#### 2.3. Co-Design

## 3. Results

^{®}, and the dynamics (1), (2), (4) were integrated numerically using the stiff ode23t solver. The optimisation problem is non-convex and partially discontinuous. The latter results from the piezoelectric kick and the stiffness of the contact model. Therefore, a genetic algorithm from the global optimisation toolbox [22] was used. See Table 2 for the fixed component and control parameters and Table 1 for an overview of the optimisation parameters and their linear bounds. The existence of both equilibrium positions is ensured by shifting the magnetic field to match the upper resting position and using a nonlinear constraint ${F}_{\mathrm{pm}}\left(0\right)\le 0.5\phantom{\rule{0.166667em}{0ex}}{F}_{\mathrm{g}}$ for the lower position. For the following study, we used the weighting coefficients given in Table 3.

^{−1}, largely reducing the necessary voltage input. Without the piezoelectric input, the optimal trajectory requires an initial voltage of 38 $\mathrm{V}$ for achieving the same transient time. This is also reflected by the trajectory costs ${J}_{\mathrm{t}}$, which are $0.284$ for the cooperative solution and $0.352$ for the simulation without the kick.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The schematic of the magnetic microactuator is shown. It consists of a movable proof mass within a guiding glass tube, a piezoelectric stack actuator (kick actuator), permanent magnets and an electromagnet (catch actuator).

**Figure 2.**Stationary magnetic force characteristics depending on the proof mass position for two different magnet setups. Both the strong magnetic field (solid, black) and the weak field (dashed, red) have stable equilibrium positions at $z=0$ and $z=2.5\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$. The dotted, grey line corresponds to the magnitude of the gravitational force.

**Figure 3.**Optimised result: (

**a**) The proof mass motion ${z}_{\mathrm{ref}}\left(t\right)$ with corresponding input ${u}_{\mathrm{ref}}\left(t\right)$, accelerated by a kick (solid, black), and an optimised trajectory without kick (dashed, red). The dotted, grey line shows the reference equilibrium over time. (

**b**) The magnetic field (solid, black), resulting from the superposition of the two single magnetic fields (dashed, red and dashed-dotted, blue), nearly fulfils the target specifications (dashed and crosses, grey). The gravitational force is illustrated by the grey, dotted line.

**Figure 4.**Flatness-based following control with model uncertainty is implemented. (

**a**) The feedback controller is mostly able to compensate the model differences, but for the largest magnetic field deviation (solid, green), the position of the proof mass briefly coincides with the solenoid centre. The dashed, red line corresponds to the reference. (

**b**) In each simulation, the magnetic force deviates more from the reference (dashed, red).

**Table 1.**Optimisation variables ${\mathit{p}}_{\mathrm{d}}$ and ${\mathit{p}}_{\mathrm{t}}$ with corresponding linear bounds used in the co-design.

Variable | Description | Lower Bound | Upper Bound | Optimised Value |
---|---|---|---|---|

${B}_{\mathrm{r},\mathrm{pm},1}$ | Ring magnet remanence | $0.1$$\mathrm{T}$ | 1 $\mathrm{T}$ | $0.2433$$\mathrm{T}$ |

${B}_{\mathrm{r},\mathrm{pm},2}$ | Ring magnet remanence | $-1$$\mathrm{T}$ | 1 $\mathrm{T}$ | $0.1425$$\mathrm{T}$ |

${B}_{\mathrm{r},\mathrm{p}}$ | Proof mass remanence | $0.01$$\mathrm{T}$ | 1 $\mathrm{T}$ | $0.9977$$\mathrm{T}$ |

${z}_{\mathrm{pm},1}$ | Position of ring magnet 1 | 1 $\mathrm{m}$$\mathrm{m}$ | 10 $\mathrm{m}$$\mathrm{m}$ | $2.773$$\mathrm{m}$$\mathrm{m}$ |

${z}_{\mathrm{pm},2}$ | Position of ring magnet 2 | $-7.5$$\mathrm{m}$$\mathrm{m}$ | 1 $\mathrm{m}$$\mathrm{m}$ | $-5.329$$\mathrm{m}$$\mathrm{m}$ |

${z}_{\mathrm{em}}$ | Position of solenoid | $2.8$$\mathrm{m}$$\mathrm{m}$ | – | $3.087$$\mathrm{m}$$\mathrm{m}$ |

${u}_{\mathrm{p}}$ | Piezo actuator voltage peak | 0$\mathrm{V}$ | ${U}_{\mathrm{max}}$ | $65.78$$\mathrm{V}$ |

${u}_{i}$ | Third motion derivative | – | – | – |

${T}_{\mathrm{kick}}$ | Controller switch on time | 0 $\mathrm{s}$ | $0.02$$\mathrm{s}$ | $3.71$$\mathrm{m}$$\mathrm{s}$ |

${T}_{\mathrm{f}}$ | Maximum transient time | $1.1\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{kick}}$ | ${T}_{\mathrm{sim}}$ | $97.12$$\mathrm{m}$$\mathrm{s}$ |

**Table 2.**Fixed design and control parameters that are used in the simulation. LQR, linear quadratic regulator.

Description | Value | Description | Value |
---|---|---|---|

Inner radius (ring magnets) | $4.5$$\mathrm{m}$$\mathrm{m}$ | ${U}_{\mathrm{max}}$ (piezo) | $100\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$ |

Outer radius (ring magnets) | $5.0$$\mathrm{m}$$\mathrm{m}$ | Stiffness (piezo) | $1.0909$$\mathrm{k}$$\mathrm{N}$$\mathrm{m}$$\mathrm{m}$^{−1} |

Height (ring magnets) | $1.5$$\mathrm{m}$$\mathrm{m}$ | Surface (piezo) | 9 $\mathrm{m}$$\mathrm{m}$^{2} |

Inner radius (solenoid) | $0.8$$\mathrm{m}$$\mathrm{m}$ | Height (piezo) | 21 $\mathrm{m}$$\mathrm{m}$ |

Outer radius (solenoid) | $2.5$$\mathrm{m}$$\mathrm{m}$ | Mass (piezo) | $1.8$$\mathrm{g}$ |

Height (solenoid) | $1.5$$\mathrm{m}$$\mathrm{m}$ | Diameter (proof mass) | $1.0$$\mathrm{m}$$\mathrm{m}$ |

Wire diameter (solenoid) | $25\mu \mathrm{m}$ | Height (proof mass) | $1.0$$\mathrm{m}$$\mathrm{m}$ |

Specific resistance (solenoid) | $18\mathrm{n}\Omega \mathrm{m}$ | Density (proof mass) | 7874 $\mathrm{k}\mathrm{g}$$\mathrm{m}$^{−3} |

Number of coils (solenoid) | 4000 | State penalty Q (LQR) | $\mathrm{diag}\left(\right[5\times {10}^{12},2\times {10}^{9},2\times {10}^{3}\left]\right)$ |

${F}_{\mathrm{max}}$ (piezo) | 360 $\mathrm{N}$ | Input penalty R (LQR) | 1 |

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

${w}_{\mathrm{d}}$ | 8 | ${w}_{\mathrm{t},1}$ | $2.22\times {10}^{4}$ |

${w}_{\mathrm{d},1}$ | 28 | ${w}_{\mathrm{t},2}$ | 0 |

${w}_{\mathrm{d},2}$ | $55\times {10}^{6}$ | ${w}_{\mathrm{t},3}$ | 0.25 |

${w}_{\mathrm{d},3}$ | $70\times {10}^{7}$ | ${w}_{\mathrm{t},4}$ | $2.22\times {10}^{7}$ |

${w}_{\mathrm{t}}$ | 1 |

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**MDPI and ACS Style**

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* **2020**, *64*, 23.
https://doi.org/10.3390/IeCAT2020-08494

**AMA Style**

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*. 2020; 64(1):23.
https://doi.org/10.3390/IeCAT2020-08494

**Chicago/Turabian Style**

Olbrich, Michael, Arwed Schütz, Koustav Kanjilal, Tamara Bechtold, Ulrike Wallrabe, and Christoph Ament.
2020. "Co-Design and Control of a Magnetic Microactuator for Freely Moving Platforms" *Proceedings* 64, no. 1: 23.
https://doi.org/10.3390/IeCAT2020-08494