# Investigation of Position and Velocity Stability of the Nanometer Resolution Linear Motor Stage with Air Bearings by Shaping of Controller Transfer Function

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- (1)
- No friction forces appear in the moving LMS parts;
- (2)
- All parts of the LMS are absolutely rigid. That is, only the rigid body dynamics is considered in the present investigation;
- (3)
- There is no backlash in the connections of the moving LMS parts;
- (4)
- The motion of the all moving LMS parts is straight-linear;
- (5)
- The displacements of all the moving LMS parts are identical;
- (6)
- The current amplifier is considered as linear after tuning of the PI controller in the current control loop;
- (7)
- The moving mass is constant and the amplifier bus voltage is constant;
- (8)
- The environmental conditions are constant and well-controlled.

## 3. Modeling and Identification

#### 3.1. Governing Equations

#### 3.2. Theoretical Transfer Functions, Their Frequency Response Functions and a Comparison with the Experimentally Identified Frequency Response Functions

## 4. Results and Discussion

- (1)
- An excitation of the system with the harmonic signal of the constant amplitude to identify the transfer functions of the systems under investigation. This investigation gives the information concerning the frequency domain.
- (2)
- Tuning of the system to adjust the velocity loop shaping filters.
- (3)
- Excitation of the moving part of LMS by different velocities.
- (4)
- Analysis of the time dependent displacement of the moving platform, see detail 1 in Figure 1, at steady dynamic process ($\ddot{x}$ = 0) as a function of the different velocities $\dot{x}\in \left\{1,5,10,20\right\}$ mm/s and different controllers: PID1, PID2 and PID3, while the mass of load ${m}_{tot}\approx 100$ kg was constant.

^{2}:

- (1)
- When the velocity $\dot{x}=1$ m/s, the LMS with controller PID2 attains both minimum values $min\left\{{m}_{{\Delta}_{e}}\right\}=\left|{m}_{{\Delta}_{e,PID2}}\right|=14.2$ nm and $min\left\{{s}_{\Delta}{}_{e}\right\}={s}_{{\Delta}_{e,PID2}}=50.6$ nm.
- (2)
- When the velocity $\dot{x}=5$ m/s, the LMS with controller PID3 attains both minimum values $min\left\{\left|{m}_{{\Delta}_{e}}\right|\right\}=\left|{m}_{{\Delta}_{e,PID3}}\right|=21.1$ nm and $min\left\{{s}_{\Delta}{}_{e}\right\}={s}_{{\Delta}_{e,PID3}}=63.3$ nm.
- (3)
- When the velocity $\dot{x}=10$ m/s, the LMS with controller PID3 attains minimum values of the estimated mean $min\left\{\left|{m}_{\Delta}{}_{e}\right|\right\}={m}_{{\Delta}_{e,PID3}}\vee 19.9$ nm, while the LMS with PID1 attains the minimum estimated standard deviation: $min\left\{{s}_{\Delta}{}_{e}\right\}={s}_{{\Delta}_{e,PID1}}=31.7$ nm.
- (4)
- When the velocity $\dot{x}=20$ m/s, the LMS with controller PID1 attains minimum values of the estimated mean $min\left\{\left|{m}_{\Delta}{}_{e}\right|\right\}=\left|{m}_{{\Delta}_{e,PID1}}\right|=66.5$ nm, while the LMS with PID3 attains the minimum estimated standard deviation: $min\left\{{s}_{\Delta}{}_{e}\right\}={s}_{{\Delta}_{e,PID3}}=85.3$ nm.

## 5. Conclusions

- (1)
- To decrease the dynamic displacement error of the platform, the order of the polynomials of the transfer function of the considered long-travel linear motor stage should be increased with the decreased velocity of the displacements of the stage;
- (2)
- The transfer functions of the fourth order polynomials can be good enough to obtain an appropriate dynamic error of the displacement of the platform when the velocity of the platform is 10 mm/s and 20 mm/s;
- (3)
- The minimums of the estimated mean and the standard deviation of the dynamic displacement error of the platform were attained with the following controllers depending on the exciting velocity:
- At 1 mm/s exciting velocity, the best controller consists of low pass and notch filters (controller PID2); the absolute value of the estimated mean and the standard deviation are 14.2 nm and 50.6 nm, respectively;
- At 5 mm/s exciting velocity, the best controller consists of only one low pass filter (controller PID1); the estimated mean and the standard deviation are 21.1 nm and 63.3 nm, respectively;
- At 10 mm/s exciting velocity, according to the estimated standard deviation of the dynamic error, the best controller consists of one low pass filter (controller PID1); the standard deviation is 66.5 nm. However, according to the estimated mean, the best controller consists of a low pass filter, notch filter and second-order bi-quad filter (controller PID3); the estimated mean is 15.9 nm.
- At 20 mm/s exciting velocity, according to the estimated standard deviation of the dynamic error, the best controller consists of a low pass filter, notch filter and second-order bi-quad filter (controller PID3); the standard deviation is 85.3 nm. However, according to the estimated mean, the best controller consists only of a low pass filter (controller PID1); the estimated mean is 31.7 nm.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BF | second order bi-quad filter; |

BLDC | brushless DC electric motor; |

EMF | electromotive force; |

FRF | frequency response function; |

LMS | long travel linear motor stage with air bearings; |

LPF | low pass filter; |

plant | LMS with current amplifier, including the current control loop consisting of proportional-integral controller; |

NF | notch filter; |

H_{OLSYS} | modelled transfer function of open loop system; |

PID | proportional-integral-derivative controller; |

OLSYS | open loop system; |

$H$ | transfer function; |

$L$ and $\widehat{L}$ | theoretically derived and experimentally identified frequency response functions; |

${\Delta}_{e}$ and ${A}_{{\Delta}_{e}}$ | dynamic error and amplitude of the dynamic error of the displacement of the stage of the long travel linear motor stage with air bearings; |

$\dot{x}$ and $\ddot{x}$ | excitation velocity and acceleration; |

$min\left\{{\Delta}_{e}\right\}$ and $max\left\{{\Delta}_{e}\right\}$ | minimum and maximum of the dynamic error of the displacement of the stage; |

${s}_{\Delta}{}_{e}$ and ${m}_{\Delta}{}_{e}$ | estimated standard deviation and mean of the dynamic error of the displacement of the stage. |

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**Figure 1.**The general view of the long travel linear motor stage with the air bearings with load (

**a**) and without load (

**b**): 1 is the moving platform; 2 is the granite base; 3 are the accelerometers; 4 is the load; 5 is the interferometer mirror; 6 are the air bearings; and 7 is the three-phase linear BLDC motor “TECNOTION UM9N.”

**Figure 2.**The general view of the test equipment: 1 is the granite base; 2 is the Michelson interferometer RENISHAW XL-80; 3 is the long travel linear motor stage with the air bearings (LMS); 4 is the PC with control and diagnostic software; 5 is the setup based on controller ACS SP + EC-04000032NAN5NDNN and motor driver (current amplifier) ACS UDMSD2B2N0N.

**Figure 4.**Frequency response functions of the plant: ${\widehat{L}}_{PLANTi}$, $i\in \left\{1,2,3\right\}$ are the experimentally identified FRF and ${L}_{PLANT}$ is the FRF of the theoretical transfer function ${H}_{PLANT}$ given in Equation (6).

**Figure 5.**Frequency response functions of controller PID1: ${\widehat{L}}_{PID1}$ is the experimentally identified FRF and ${L}_{PID1}$ is the FRF of the theoretical transfer function ${H}_{PID1}$ given in Equation (7).

**Figure 6.**Frequency response functions of controller PID2: ${\widehat{L}}_{PID2}$ is the experimentally identified FRF and ${L}_{PID2}$ is the FRF of the theoretical transfer function ${H}_{PID2}$ given in Equation (8).

**Figure 7.**Frequency response functions of controller PID3: ${\widehat{L}}_{PID3}$ is the experimentally identified FRF and ${L}_{PID3}$ is the FRF of the theoretical transfer function ${H}_{PID3}$ given in Equation (9).

**Figure 8.**Experimentally and analytically obtained FRFs, ${\widehat{L}}_{OLSYS1}$ and ${L}_{OLSYS1}$, respectively, of LMS with controller PID2.

**Figure 9.**Experimentally and analytically obtained FRFs, ${\widehat{L}}_{OLSYS2}$ and ${L}_{OLSYS2}$, respectively, of LMS with controller PID2.

**Figure 10.**Experimentally and analytically obtained FRFs, ${\widehat{L}}_{OLSYS3}$ and ${L}_{OLSYS3}$, respectively, of LMS with controller PID3.

**Figure 11.**Dynamic displacement errors ${\Delta}_{e}$ of the LMS platform with different controllers PID1, PID2 and PID3 at different velocities $\dot{x}\in \left\{1,5,10,20\right\}$ mm/s and at the constant acceleration $\ddot{x}=0$.

**Table 1.**Summarized parameters of the transfer functions ${H}_{PID1}$, ${H}_{PID2}$ and ${H}_{PID3}$.

Parameter | ${\mathit{H}}_{\mathit{P}\mathit{I}\mathit{D}1}$ | ${\mathit{H}}_{\mathit{P}\mathit{I}\mathit{D}2}$ | ${\mathit{H}}_{\mathit{P}\mathit{I}\mathit{D}3}$ |
---|---|---|---|

LPF cut off frequency, Hz | 45 | 98 | 150 |

NF central frequency, Hz | - | 45 | 110 |

BF central frequency, Hz | - | - | 45 |

BF type | - | - | Notch filter |

Position loop P coefficient | 5 | 5 | 5 |

Velocity loop P coefficient | 200 | 180 | 180 |

Velocity loop I coefficient | 120 | 160 | 160 |

Controller | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

PID1 | PID2 | PID3 | ||||||||||

Estimations of the dynamic displacement error, ${\Delta}_{e}$ in nm | ||||||||||||

Velocity, mm/s | min{Δ_{e}} | max{Δ_{e}} | ${s}_{\Delta}{}_{e}$ | ${m}_{\Delta}{}_{e}$ | min{Δ_{e}} | max{Δ_{e}} | ${s}_{\Delta}{}_{e}$ | ${m}_{\Delta}{}_{e}$ | min{Δ_{e}} | max{Δ_{e}} | ${s}_{\Delta}{}_{e}$ | ${m}_{\Delta}{}_{e}$ |

1 | −240.00 | 240.00 | 79.70 | 58.20 | −160.00 | 160.00 | 50.60 | −14.20 | −240.00 | 160.00 | 65.10 | −79.03 |

5 | −400.00 | 560.00 | 213.00 | 51.80 | −400.00 | 320.00 | 140.00 | −37.50 | −160.00 | 240.00 | 63.30 | 21.10 |

10 | −160.00 | 160.00 | 66.50 | −30.50 | −400.00 | 320.00 | 130.00 | 26.70 | −320.00 | 320.00 | 122.00 | 15.90 |

20 | −240.00 | 320.00 | 117.00 | 31.70 | −160.00 | 320.00 | 92.10 | 75.80 | −320.00 | 160.00 | 85.30 | −62.70 |

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## Share and Cite

**MDPI and ACS Style**

Piščalov, A.; Urbonas, E.; Višniakov, N.; Zabulionis, D.; Kilikevičius, A.
Investigation of Position and Velocity Stability of the Nanometer Resolution Linear Motor Stage with Air Bearings by Shaping of Controller Transfer Function. *Symmetry* **2020**, *12*, 2062.
https://doi.org/10.3390/sym12122062

**AMA Style**

Piščalov A, Urbonas E, Višniakov N, Zabulionis D, Kilikevičius A.
Investigation of Position and Velocity Stability of the Nanometer Resolution Linear Motor Stage with Air Bearings by Shaping of Controller Transfer Function. *Symmetry*. 2020; 12(12):2062.
https://doi.org/10.3390/sym12122062

**Chicago/Turabian Style**

Piščalov, Artur, Edgaras Urbonas, Nikolaj Višniakov, Darius Zabulionis, and Artūras Kilikevičius.
2020. "Investigation of Position and Velocity Stability of the Nanometer Resolution Linear Motor Stage with Air Bearings by Shaping of Controller Transfer Function" *Symmetry* 12, no. 12: 2062.
https://doi.org/10.3390/sym12122062