# Design and Optimal Control of a Multistable, Cooperative Microactuator

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Modeling

^{®}Maxwell [24] and are implemented as spline-interpolated lookup tables.

#### 2.2. Trajectory Planning and Control

#### 2.2.1. Piezoelectric Kick

#### 2.2.2. Flatness-Based Control

#### 2.2.3. Optimal Trajectory Planning

## 3. Results

^{®}[31], with the exception of the magnetic forces, which were computed with ANSYS

^{®}Maxwell [24]. The optimization problem was solved using CasADi [32] with the solver IPOPT [33].

#### 3.1. Multistability Consideration

#### 3.2. Simulation Study

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The multistable microactuator consists of a movable, cylindrical magnet within a guiding glass tube, a piezoelectric stack actuator (kick actuator), permanent magnets and electromagnets (catch actuators). The transparent part depicts a possible extension to a third levitating equilibrium.

**Figure 2.**Combination of cascade control and flatness-based control. The linear partial system, i.e., the solenoids, are controlled by linear state feedback with controller gain matrix ${K}_{\mathrm{i}}$ and prefilter ${S}_{\mathrm{i}}$ (

**a**) and the nonlinear system is handled using a feedback linearization approach (

**b**). A trajectory generator is used to compute optimal reference motions and deviations between the current state and reference are compensated by a second linear controller with controller gain ${K}_{\mathrm{z}}$ (

**c**).

**Figure 3.**Superimposed proof mass trajectory that fulfills the initial and final condition up to the second order. A nominal trajectory (blue) and an optimized, piecewise constant function (red) are superimposed and result in an optimal trajectory (black). The amplitudes of the piecewise function correspond to the optimization variables $\sigma $.

**Figure 4.**Simulated magnetic force characteristic ${F}_{\mathrm{pm}}$ of the superimposed magnetic fields. The intersections with the gravitational force (dashed, gray) correspond to the predefined, stable equilibria and are marked by red circles. (

**a**) Setup 1: two equilibria are generated by superimposing two magnetic fields. (

**b**) Setup 2: the identical resting positions are achieved by three ring magnets, but the choice of magnet parameters is more flexible. The third equilibrium of Setup 2, which is above 7 mm, is unintended and therefore not shown.

**Figure 5.**Magnetic force characteristic for the extended, multistable microactuator. (

**a**) Setup 3: Three stable equilibria (red circles) were generated by four ring magnets. (

**b**) Setup 4: Four resting positions could be achieved with five ring magnets.

**Figure 6.**Alternative concepts for achieving suitable magnetic force characteristics.(

**a**) A Halbach array-like magnet design may be used to weaken the magnetic field. (

**b**) Multiple single magnets can be used instead of a small ring magnet.

**Figure 7.**Combined kick and catch trajectories to both equilibria for different piezoelectric kick voltages between 0 $\mathrm{V}$ and 100 $\mathrm{V}$. Top panel: position z of the proof mass. Bottom panel: the respective sum of squared electrical currents. (

**a**) The kick to the equilibrium at 2.5 mm is shown. (

**b**) A similar effect is achieved for the kick to the equilibrium at 5 mm. In both cases, the electrical currents are reduced with increasing kick impact.

**Figure 8.**Feedback control simulation compensating for initial state uncertainty. The expected initial state resulting in the reference trajectory (dashed, black) for 50 $\mathrm{V}$ is not achieved due to incorrect kick voltages. Feedback control compensates this error and the proof mass motion asymptotically converges to the desired trajectory for different piezo voltages.

**Figure 9.**The trajectory costs over the number of parameters are shown. The red circle corresponds to the polynomial trajectory and the blue circles to the optimized motions. Although there is a fast cost function decrease for a low number of parameters, the costs oscillate and slightly increase for more than 10 optimization variables.

**Figure 10.**Comparison of trajectories and respective currents with different numbers of optimization variables. (

**a**) Top panel: for the polynomial trajectory, smooth motions were generated and the transition times were optimized. Middle panel: the trajectory with two parameters ${\sigma}_{s}$ and optimized transient time is less smooth but has lower costs. Bottom panel: the lowest costs were achieved with 20 parameters ${\sigma}_{s}$. The dashed line corresponds to the desired final position of the proof mass. (

**b**) The sum of squared currents corresponding to the different trajectories is shown. For the trajectory with 20 parameters, the solution is non-smooth due to fast changes in the current, but the transient time and the sum of squared currents are the lowest.

Component | Height in mm | Inner Radius in mm | Outer Radius in mm | Mass in mg |
---|---|---|---|---|

Proof mass | 1 | – | $0.5$ | $3.77$ |

Ring magnet | $0.5$ | $1.5$ | $2.2$ | – |

**Table 2.**Experimental Setups 1–4 and respective parameters used to generate multiple equilibrium positions.

Setup | Magnet | Remanence in T | Position in mm |
---|---|---|---|

1 | ${B}_{\mathrm{r},\mathrm{p}}$ | 0.1 | – |

${B}_{\mathrm{r},\mathrm{pm},1}$ | 0.12 | 1.592 | |

${B}_{\mathrm{r},\mathrm{pm},2}$ | 0.1 | 3.296 | |

2 | ${B}_{\mathrm{r},\mathrm{p}}$ | 0.2 | – |

${B}_{\mathrm{r},\mathrm{pm},1}$ | 0.2 | 1.225 | |

${B}_{\mathrm{r},\mathrm{pm},2}$ | 0.2 | 3.746 | |

${B}_{\mathrm{r},\mathrm{pm},3}$ | 0.2 | 6.338 | |

3 | ${B}_{\mathrm{r},\mathrm{p}}$ | 0.2 | – |

${B}_{\mathrm{r},\mathrm{pm},1}$ | 0.2 | 1.143 | |

${B}_{\mathrm{r},\mathrm{pm},2}$ | 0.2 | 3.807 | |

${B}_{\mathrm{r},\mathrm{pm},3}$ | 0.2 | 6.223 | |

${B}_{\mathrm{r},\mathrm{pm},3}$ | 0.2 | 8.868 | |

4 | ${B}_{\mathrm{r},\mathrm{p}}$ | 0.2 | – |

${B}_{\mathrm{r},\mathrm{pm},1}$ | 0.2 | 1.283 | |

${B}_{\mathrm{r},\mathrm{pm},2}$ | 0.2 | 3.670 | |

${B}_{\mathrm{r},\mathrm{pm},3}$ | 0.2 | 6.344 | |

${B}_{\mathrm{r},\mathrm{pm},4}$ | 0.2 | 8.701 | |

${B}_{\mathrm{r},\mathrm{pm},5}$ | 0.2 | 11.42 |

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

Inner radius (solenoid) | 0.8 mm | Resistance (solenoid) | 281 Ω |

Outer radius (solenoid) | 1.2 mm | ${F}_{\mathrm{max}}$ (piezo) | 360 N |

Height (solenoid) | 1.5 mm | ${U}_{\mathrm{max}}$ (piezo) | 100 V |

Wire diameter (solenoid) | 25 μm | ${k}_{\mathrm{A}}$ (piezo) | 1.1 MNm^{−1} |

Specific resistance (solenoid) | 18 nΩm | M (piezo) | 1.8 g |

Number of turns (solenoid) | 1222 | ${c}_{\mathrm{A}}$ (piezo) | 25 |

Inductance (solenoid) | 3.9 mH | $\mathrm{f}$ (proof mass) | 2.5 × 10^{−4} |

Mutual inductance (solenoid) | 2.0 mH | ${k}_{\mathrm{c}}$ (contact) | 4.5 × 10^{6} |

Position (solenoid 1) | 2.2 mm | ${F}^{\ast}$ (contact) | $m\phantom{\rule{0.166667em}{0ex}}g$ |

Position (solenoid 2) | 4.7 mm | ${r}_{\mathrm{d}}$ (contact) | 3 |

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

${\mathit{p}}_{\mathrm{i}}$ | $[-400+100\mathrm{i},-400-100\mathrm{i}]$ |

${\mathit{p}}_{\mathrm{z}}$ | $[-200+100\mathrm{i},-200-100\mathrm{i},-150]$ |

${\mathit{K}}_{\mathrm{i}}$ | $[-280.3,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1.19;\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0.41,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}-279.8]$ |

${\mathit{K}}_{\mathrm{z}}$ | $[7500,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}110,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0.5]\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{3}$ |

${T}_{\mathrm{kick}}$ | 1 ms |

$\beta $ | 1 |

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**MDPI and ACS Style**

Olbrich, M.; Schütz, A.; Bechtold, T.; Ament, C.
Design and Optimal Control of a Multistable, Cooperative Microactuator. *Actuators* **2021**, *10*, 183.
https://doi.org/10.3390/act10080183

**AMA Style**

Olbrich M, Schütz A, Bechtold T, Ament C.
Design and Optimal Control of a Multistable, Cooperative Microactuator. *Actuators*. 2021; 10(8):183.
https://doi.org/10.3390/act10080183

**Chicago/Turabian Style**

Olbrich, Michael, Arwed Schütz, Tamara Bechtold, and Christoph Ament.
2021. "Design and Optimal Control of a Multistable, Cooperative Microactuator" *Actuators* 10, no. 8: 183.
https://doi.org/10.3390/act10080183