# Reaction Force-Based Position Sensing for Magnetic Levitation Platform with Exceptionally Large Hovering Distance

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Magnetic Levitation Platform Overview

## 3. Dynamics Modeling

#### 3.1. Radial Motion Dynamics

#### 3.2. Rotational Dynamics

#### 3.3. Force Sensor Mechanical Dynamics

#### 3.4. Force Sensor Electrical Dynamics

#### 3.5. Summary of the Dynamics

## 4. Dynamic Model Verification and Tuning

#### 4.1. Mover’s Position Controller

#### 4.2. Dynamic Model Proof and Adaption

^{®}and adding the obtained frequency response to the model transfer function ${G}_{\mathrm{tot},\mathrm{x}}$, the measurement performed with the mover levitating can be reproduced with better accuracy (cf. ${G}_{\mathrm{meas},\mathrm{x}}$ with ${G}_{\mathrm{tot},\mathrm{x}}+{G}_{\mathrm{dist},\mathrm{xx}}$ in Figure 7c). The error that remains between the transfer functions can be attributed to the fact that the two measurements (with and without the mover) are performed separately, and the model parameters also contain errors originating from their measurement and/or simulation. Another behavior that is hard to model but is prominent in the output of the force sensor, as shown in Figure 7d, is the cross-coupling between the axes, which is represented by the transfer function ${G}_{\mathrm{dist},\mathrm{yx}}$ for the force sensor’s output ${u}_{\mathrm{out},\mathrm{x}}$ due to a current ${I}_{\mathrm{y}}$ when the system is excited without the mover. For this measurement, the same disturbances as those observed for ${G}_{\mathrm{dist},\mathrm{xx}}$ are visible, where the movement of the stator and EMs in the y direction causes a false reading in the x direction, and vice versa. In this measurement, the amplitude of the disturbance on the sensor output is slightly above the noise level for the lower frequency range. Between 10 Hz and 25 Hz, only noise is measured, as can be seen in the distribution of the measurement points, especially in the phase plot.

## 5. Observer and Controller Design

#### 5.1. Observer

#### 5.2. Mover’s Position Controller and Rotation Damping

#### 5.3. Inverter Stage

^{®}function “pidTuner”, where the closed-loop bandwidth ${\omega}_{\mathrm{in}}$, given in Table 3, of the inner current control loop is selected according to the transfer function of the designed filter from the inverter voltage to the inverter current ${G}_{\mathrm{el}}\left(s\right)={I}_{\mathrm{inv}}\left(s\right)/{U}_{\mathrm{inv}}\left(s\right)$. As indicated in Figure 8, a filter ${G}_{\mathrm{filt}}\left(s\right)$ on the feedback path of the inner current controller is used to better approximate the current in the EM $I\left(s\right)={G}_{\mathrm{LC},\mathrm{EM}}\left(s\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{U}_{\mathrm{inv}}\left(s\right)$ from the measured current ${I}_{\mathrm{inv}}\left(s\right)$. To minimize the phase shift, its cutoff frequency should be larger than the desired bandwidth ${\omega}_{\mathrm{in}}$. For the controller design, the filter for the current is included in the open-loop transfer function ${G}_{\mathrm{el}}\left(s\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{G}_{\mathrm{filt}}\left(s\right)$, and care is taken to achieve a sufficient phase margin that avoids overshoots in the EMs’ current.

#### 5.4. Position Sensor Offset Compensation

## 6. Hardware Demonstrator Realization

#### 6.1. Force Sensor

#### 6.2. Controller Tuning

#### 6.3. Results

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Design of the Electromagnets

Design Parameters | |||

${U}_{\mathrm{max}}$ | 40 V | ${I}_{\mathrm{max}}$ | 6 A |

${x}_{\mathrm{m}}$ | 10 mm | ${w}_{\mathrm{EM}}$ | $14.9$ mm |

${d}_{\mathrm{w},\mathrm{Cu}}$ | $0.75$ mm | ${h}_{\mathrm{EM}}$ | 23 mm |

${d}_{\mathrm{w}}$ | $0.82$ mm | ||

Calculated Characteristics | |||

N | 550 | ${N}_{\mathrm{hor}}$ | 20 |

${N}_{\mathrm{vert}}$ | $27.5$ | ${k}_{\mathrm{fill}}$ | $0.7$ |

${R}_{\mathrm{EM}}$ | 5.5 Ω | ${m}_{\mathrm{EM}}$ | $0.56$ kg |

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**Figure 1.**Three-dimensional rendering of the magnetic levitation platform (MLP) considered in this paper, with a reaction force sensor used to determine the mover’s $x,y$ position. The system is actively controlled with the help of electromagnets (EMs) driven by a power converter, which is controlled by a system-on-a-chip (SoC), where an observer-based position controller (cf. Section 5) is implemented.

**Figure 2.**(

**a**) Section view of a radial displacement of the mover under destabilizing magnetic forces without active control. (

**b**) The forces that the mover experiences can be measured as reaction forces on the stator and converted using a linear equation to extract the mover’s position for active position control. For the sake of illustration, the EMs’ return current path (see Figure 1) is not shown, and the levitation height is shown reduced compared to the other dimensions, which are drawn to scale.

**Figure 3.**(

**a**) Section view of the MLP with the mover in the center, ${x}_{\mathrm{m}}={y}_{\mathrm{m}}=0$, and the force/torque vectors acting on the different parts of the MLP when a current I is injected into the EMs. (

**b**–

**g**) Linearizations of different simulations and/or measurements performed on the MLP according to Table 1, aimed at building a model in the neighborhood of ${x}_{\mathrm{m}}=0$, ${y}_{\mathrm{m}}=0$, and ${\theta}_{\mathrm{m}}=0$.

**Figure 4.**Complete block diagram of the MLP, with the dynamics of the subsystems represented as Bode diagrams: (

**a**) mover tilting and/or rotation, (

**b**) mover displacement, (

**c**) force sensor’s sensing element mechanical characteristic, and (

**d**) force sensor mechanical-to-electrical signal conversion, amplification, and filtering. Parameter values are specified in Table 1.

**Figure 5.**Bode diagrams of the model for the MLP (see Figure 4). The inset shows the location of the seven poles of the system in the complex plane. A single pole is in the RHP and corresponds to the radial displacement of the mover (see (2)). The poles describing the mover’s rotation and the force sensor’s dynamics are stable but show a relatively large imaginary part, implying oscillations in the system. The magnitude and frequency axes are presented on a logarithmic scale.

**Figure 6.**Block diagram of the mover’s position control built with a PD controller that stabilizes the unstable radial dynamics and a reference tracking PI controller.

**Figure 7.**(

**a**) Top view of the measurement setup to verify the MLP’s dynamic model in the x direction. The mover’s position is controlled based on optical position measurement, and sinusoidal currents are injected into the EMs to cause a displacement of the mover. Consequently, a reaction on the stator is measured using the force sensor while the y position is kept constant at ${y}_{\mathrm{m}}=0$ with sliding barriers. (

**b**) Block diagram of the control system depicted in Figure 6 that levitates the mover. Currents ${I}_{\mathrm{exc}}$ are summed to the control currents to verify ${G}_{\mathrm{tot},\mathrm{x}}$ from Figure 5 by measuring ${G}_{\mathrm{meas},\mathrm{x}}$. (

**c**) Frequency responses of the model, measurement, and disturbance-corrected output of the force sensor along the x axis in response to an input current in the x direction (with mover levitating). (

**d**) Model disturbance ${G}_{\mathrm{dist},\mathrm{xx}}$ and cross-coupling between the y-axis current and the x-axis sensor output ${G}_{\mathrm{dist},\mathrm{yx}}$ (without the mover). Please note that the measured phase lies within $[-\pi ,\pi ]$ but has been unwrapped for this representation (i.e., adjusted by adding or subtracting $2\pi $ to targeted phase values to ensure a continuous and smooth representation without discontinuities).

**Figure 8.**Block diagram of the MLP with the mover’s dynamics observer, position, and current controllers for the x axis. The black boxes represent the ideal MLP as presented up to and including Section 3, whereas the red boxes are required to extend the model for the realized prototype discussed in Section 4.2 so that the observer delivers the states as close as possible to reality for the position controller. For these disturbance corrections, the y-axis current plays a role in the x-axis model, and vice versa.

**Figure 9.**Electrical system consisting of a full-bridge inverter driving the EM of an axis. A passively damped $LC$ filter is inserted to reduce the disturbance ${i}_{\mathrm{HF},2}$, which can affect the position measurement circuitry due to coupling denoted by ${C}_{\mathrm{c}}$ with the EM, modeled by four lumped elements. Similarly, the common-mode choke helps reduce the amplitude of the high-frequency common-mode current ${i}_{\mathrm{CM}}$.

**Figure 10.**(

**a**) Excitation of the strain gauges, glued to the force sensor for a single axis, with a buffered constant-voltage reference. (

**b**) Schematic circuit diagram of the force sensor amplifier for a single axis, consisting of a manual offset correction stage, variable-gain amplification, and analog-to-digital conversion.

**Figure 11.**(

**a**) The complete system showing the mover’s stable levitation above the stator with a 104 mm air gap. The force sensor is not visible because it is placed underneath the stator and is connected to the VGA that amplifies and filters the force-dependent signals and delivers them to the observer implemented on the SoC. The DC-link supply is used for the EMs, whereas the auxiliary supply provides power to the SoC and the force sensor. (

**b**,

**c**) show the 2D plots of the observed mover’s position and control current, respectively, which are also found in (

**a**) on the oscilloscope with a different scaling factor.

Mover PM external radius | ${r}_{\mathrm{m},\mathrm{ext}}$ | $64.6$ mm |

Mover PM internal radius | ${r}_{\mathrm{m},\mathrm{int}}$ | $35.9$ mm |

Mover height | ${h}_{\mathrm{m}}$ | 5 mm |

Stator PM external radius | ${r}_{\mathrm{s},\mathrm{ext}}$ | 50 mm |

Stator PM internal radius | ${r}_{\mathrm{s},\mathrm{int}}$ | $26.5$ mm |

Stator height | ${h}_{\mathrm{s}}$ | 20 mm |

Mover weight | ${m}_{\mathrm{m}}$ | $0.36$ kg |

Stator weight | ${m}_{\mathrm{s}}$ | $0.91$ kg |

Mover moment of inertia | ${J}_{\mathrm{m}}$ | $0.58$
gm^{2} |

Total MLP weight | ${m}_{\mathrm{mlp}}$ | $3.6$ kg |

Levitation height | h | 104 mm |

Characteristic dimension | CD | 207 mm |

Radial stiffness | ${k}_{\mathrm{FPM}}$ | $32.8$ N/m |

Displacement torque const. | ${k}_{\mathrm{Tdisp}}$ | $0.25$ Nm/m |

Rotational stiffness | ${k}_{\mathrm{TPM}}$ | $1.6$ mNm/$\xb0$ |

Rotational damping | ${k}_{\mathrm{d},\mathrm{rot}}$ | 2 $\mathsf{\mu}$Nms/$\xb0$ |

Rotational force const. | ${k}_{\mathrm{Frot}}$ | $4.4$ mN/$\xb0$ |

EMs force const. | ${k}_{\mathrm{FEM}}$ | 65 mN/A |

EMs torque const. | ${k}_{\mathrm{TEM}}$ | $0.93$ mNm/A |

Force sensor damping | ${k}_{\mathrm{d},\mathrm{RFS}}$ | $0.04$ Ns/m |

Force sensor stiffness | ${k}_{\mathrm{s},\mathrm{RFS}}$ | 694 kN/m |

Force sensor el. conv. const. | ${k}_{\mathrm{v}}$ | $13.3$ $\mathsf{\mu}$V/$\mathsf{\mu}$m |

Force sensor amplifier gain | ${k}_{\mathrm{VGA}}$ | 10 V/mV |

Inverter | |||

${U}_{\mathrm{DC}}$ | 40 $\mathrm{V}$ | ${f}_{\mathrm{sw}}$ | 100 $\mathrm{k}$$\mathrm{Hz}$ |

$LC$ filter | |||

${L}_{\mathrm{f}}$ | 22 $\mathsf{\mu}$$\mathrm{H}$ | ${R}_{\mathrm{fd}}$ | 1 $\mathsf{\Omega}$ |

${C}_{\mathrm{f}}$ | 22 $\mathsf{\mu}$$\mathrm{F}$ | ${C}_{\mathrm{fd}}$ | 88 $\mathsf{\mu}$$\mathrm{F}$ |

${f}_{0,\mathrm{LC}}$ | $7.2$ $\mathrm{k}$$\mathrm{Hz}$ | n | 4 |

Electromagnet | |||

${L}_{\mathrm{EM}}$ | $18.6$ $\mathrm{m}$$\mathrm{H}$ | ${R}_{\mathrm{EM}}$ | $5.5$ $\mathsf{\Omega}$ |

${C}_{\mathrm{EM}}$ | 87 $\mathrm{p}$$\mathrm{F}$ | ${R}_{\mathrm{EMd}}$ | 85 $\mathrm{k}$$\mathsf{\Omega}$ |

Resonant freq. | 125 $\mathrm{k}$$\mathrm{Hz}$ |

**Table 3.**Parameters of the outer current controller, position and active rotation damping controller, and inner current controller.

Outer Current Controller ${G}_{\mathrm{I},\mathrm{out}}$ | |||

${\omega}_{\mathrm{out}}$ | $2\pi \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}0.05\mathrm{Hz}$ | ||

Position controller (50) | |||

${k}_{1}$ | $-0.04$ As/$\xb0$ | ${k}_{2}$ | $-4.3$ A/$\xb0$ |

${k}_{3}$ | $283.5$ As/m | ${k}_{4}$ | $3.09$ kA/m |

H | $536.8$ A/m | ${k}_{\mathrm{I}}$ | $2\pi \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}0.15\mathrm{A}/\mathrm{ms}$ |

${p}_{1}$ | $2\pi \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}0.52\mathrm{Hz}$ | ${p}_{2}$ | $2\pi \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}3.29\mathrm{Hz}$ |

${p}_{3}$ | $2\pi \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}3.18\mathrm{Hz}$ | ${p}_{4}$ | $2\pi \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}1.74\mathrm{Hz}$ |

Inner current controller ${G}_{\mathrm{PI},\mathrm{in}}$ | |||

${\omega}_{\mathrm{in}}$ | $2\pi \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}20\mathrm{Hz}$ |

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**MDPI and ACS Style**

Bonetti, R.; Beglinger, L.; Mirić, S.; Bortis, D.; Kolar, J.W.
Reaction Force-Based Position Sensing for Magnetic Levitation Platform with Exceptionally Large Hovering Distance. *Actuators* **2024**, *13*, 114.
https://doi.org/10.3390/act13030114

**AMA Style**

Bonetti R, Beglinger L, Mirić S, Bortis D, Kolar JW.
Reaction Force-Based Position Sensing for Magnetic Levitation Platform with Exceptionally Large Hovering Distance. *Actuators*. 2024; 13(3):114.
https://doi.org/10.3390/act13030114

**Chicago/Turabian Style**

Bonetti, Reto, Lars Beglinger, Spasoje Mirić, Dominik Bortis, and Johann W. Kolar.
2024. "Reaction Force-Based Position Sensing for Magnetic Levitation Platform with Exceptionally Large Hovering Distance" *Actuators* 13, no. 3: 114.
https://doi.org/10.3390/act13030114