# Development of Air Bearing Stage Using Flexure for Yaw Motion Compensation

^{1}

^{2}

^{*}

## Abstract

**:**

_{z}directions. To work with Θ

_{z}as the rotational motion of the stage, we applied a flexure consisting of four bar linkages. The stage from a previous study in which flexure is applied to compensate for yaw motion error has the limitation of increasing the structural stiffness of the stage due to the rotational stiffness. In this study, we propose a combination of a new stage structure and flexure to ensure the high structural stiffness of the stage and the very low rotational stiffness of the flexure at the same time. Modeling and design optimization were performed to apply adequate flexure to the proposed stage. Experiments were carried out to verify yaw motion error compensation and the performance of the stage. The proposed stage has a maximum yaw motion error of 0.86 arcsec during the scanning motion and 48 ms settling time, while the stepping motion is improved by 34.2% compared to the previous study.

## 1. Introduction

_{z}). Yaw motion error is a very important problem because it largely affects the final performance of the stage. Even a very small error should be eliminated, as it can lead to an exaggerated large disagreement on the outer edge of the workpiece. Nonetheless, efforts to remove the causes of yaw motion error at the level of fabrication or assembly are limited.

## 2. Design of Air Bearing Stage

#### 2.1. Stacked Gantry Structure

#### 2.2. Air Bearings and Flexure

_{f}denotes the distance from the center of the crossbeam to the center of the flexure and is 383.75 mm. Θ is the yaw motion of the crossbeam. As seen from (1) and (2), the flexure eliminates the risk of damage to the pads and the guide surface or functional failure of the air bearings when compensating for the yaw motion error.

#### 2.3. Linear Motor and Flexure

## 3. Design of Flexure

#### 3.1. Conceptual Design

#### 3.2. Mathematical Modeling

_{x}, I

_{y}, and I

_{z}are the moments of inertia with respect to local coordinate systems, as shown in (6). Since the number of links of one flexure is 17, M and K are 102 × 102 matrices, as shown in (5) and (7). Each element of (8) is the stiffness of each notch hinge in global coordinates. T

_{k}

^{i}is the transformation matrix to change the coordinates from local to global. F is the force vector indicating all force components exerted on each link, as shown in (9). The displacement vector x is defined by (10), and qi is the displacement vector of the center point of the ith link, which can be expressed by (11).

_{x}. The stiffness in other directions, K

_{y}, K

_{z}, K

_{θx}, K

_{θy}, and K

_{θz}, can be obtained in the same manner.

#### 3.3. Optimization

^{2}, a reaction force of 45 N is exerted on each flexure. The stiffness of the flexure in the x-direction is constrained to 900 N/μm, which corresponds to deformation of 100 nm. Finally, the size of the flexure should be smaller than 200 × 200 × 28 mm

^{3}, and there should be no interference between the design parameters. The size is determined by the area of the Y-slider.

## 4. Implementation of Stage

#### 4.1. Manufacturing

#### 4.2. Experimental Setup

_{1}to m

_{3}, are transformed into the position feedback signals x

^{m}, y

^{m}, and θ

_{z}

^{m}using the sensor transformation matrix, as shown in Equation (14). The sensor transformation matrix is determined by the arrangement of the linear encoders. D is the distance between Y1 and Y2 encoders, as shown in Figure 6b. The position feedback signals represent the motion of the workpiece carrying the X-slider in the Cartesian coordinate system, which corresponds to the control axes. The proportional-integral-differential (PID) algorithm was used for each control axis, with the input signals to the linear motors determined from the control signals using simple kinematics, as shown in Equation (15). It was reported by Giam et al. [12] and Tan et al. [19] that the cross-coupled control scheme augmented by sliding mode control or a disturbance observer showed enhanced performance compared to conventional cross-coupled control adopting a PID algorithm. In addition, PID control schemes combined with a fuzzy controller [27] or an adaptive jerk controller [28] have been studied. However, such advanced algorithms are not easy to implement in the commercial low-level controllers generally used for wafer or display manufacturing devices. Since the air bearing stage in this study targeted the lithography process of flat panel displays, the developed system was evaluated by the cross-coupled control scheme with a simple PID algorithm. Equations (14) and (15) represent the cross-coupled control scheme. The feedback of two motors, m

_{2}and m

_{3}, are used to create two control axes, Y and Θ

_{z}. The corresponding feedback signals are y

^{m}and θ

_{z}

^{m}, which are the sum and the difference after being divided by D of two feedback signals, respectively. The trajectory command, y

^{cmd}and θ

_{z}

^{cmd}, are generated directly for the two control axes, Y and Θ

_{z}. The PID control output is then split into two components, u

_{y1}and u

_{y2}, one for each motor.

#### 4.3. Results

_{z}directions, respectively. Due to the stack structure of the stage, the motion of the X-slider is affected by the motion of the Y-sliders. Therefore, the in-position stability in the X-axis was slightly higher than that of the Y-axis. Based on the experimental results, the stage was found to have high precision in static conditions.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of stages for yaw motion error compensation: (

**a**) 3D drawing of the conventional H-type stage of the previous study [15], (

**b**) schematic front view of the conventional H-type stage, (

**c**) 3D drawing of the newly proposed stage of stack and gantry structure, and (

**d**) schematic front view of the newly proposed stage.

**Figure 2.**Change in gap between the air bearing pad and guide surface when the yaw motion error is compensated; (

**a**) air bearing stage without a flexure; (

**b**) air bearing stage with flexures.

**Figure 4.**(

**a**) Skeleton diagram (left) and 3D conceptual design (right) of the flexure for one rotational DOF motion; (

**b**) design parameters of the proposed flexure.

**Figure 5.**(

**a**) A manufactured flexure to be inserted between the crossbeam and the Y-slider; (

**b**) air bearing stage where the proposed flexures are installed.

**Figure 6.**(

**a**) Control algorithm for linear motion and yaw motion of the stage; (

**b**) setup of linear motors and linear encoders.

**Figure 7.**Experimental results of motion range and corresponding speed along (

**a**) X-axis and (

**b**) Y-axis.

**Figure 8.**Experimental results of high-precision in-position stability: (

**a**) X-axis, (

**b**) Y-axis, and (

**c**) Θ

_{z}-axis.

**Figure 9.**Experimental results of yaw motion error compensation during the scanning motion along the Y-axis, (

**a**) position and error of the stage along the Y-axis, (

**b**) position error of the stage along the X-axis and Θ

_{z}-axis, and (

**c**) electric current provided to the linear motors.

Unit | Specification | |
---|---|---|

Motion range | mm | 200 |

In-position stability | μm | ±0.05 |

Yaw motion error | arcsec | ±1.0 |

Maximum speed | mm/s | 100 |

Maximum acceleration | m/s^{2} | 2 |

Settling time 2 μm move, 1% (20 nm) | msec | 500 |

Notation | Compliance |
---|---|

c_{1} | $\left(9\pi {r}^{\frac{5}{2}}\right)/\left(2Eb{t}^{\frac{5}{2}}\right)+\left(3\pi {r}^{\frac{3}{2}}\right)/\left(2Eb{t}^{\frac{3}{2}}\right)$ |

c_{2} | $\left(12\pi {r}^{2}\right)/\left(E{b}^{3}\right)\left\{{\left(r/t\right)}^{\frac{1}{2}}-1/4\right\}$ |

c_{3} | $\left(9\pi {r}^{\frac{3}{2}}\right)/\left(2Eb{t}^{\frac{5}{2}}\right)$ |

c_{4} | $\left(12r\right)/\left(E{b}^{3}\right)\left\{\pi {\left(r/t\right)}^{\frac{1}{2}}-\left(2+\pi \right)/2\right\}$ |

c_{5} | $1/\left(Eb\right)\left\{\pi {\left(r/t\right)}^{\frac{1}{2}}-\pi /2\right\}$ |

c_{6} | $\left(12/E{b}^{3}\right)\left\{\pi {\left(r/t\right)}^{\frac{1}{2}}-\left(2+\pi \right)/2\right\}$ |

c_{7} | $\left(9\pi {r}^{\frac{1}{2}}\right)/\left(2Eb{t}^{\frac{5}{2}}\right)$ |

c_{8} | $\left(9\pi {r}^{\frac{1}{2}}\right)/\left(4Gb{t}^{\frac{5}{2}}\right)$ |

Parameters | Design Results (mm) |
---|---|

u | 19.0 |

v | 48.8 |

d | 63.4 |

e | 23.0 |

f | 10.5 |

r | 3.2 |

t | 0.6 |

Stiffness | Modeling | FEM | Error (%) |
---|---|---|---|

X (MN/m) | 976 | 996 | 2.01 |

Y (MN/m) | 976 | 949 | 2.85 |

Z (MN/m) | 492 | 523 | 5.93 |

Θ_{x} (kNm/rad) | 1360 | 1453 | 6.4 |

Θ_{y} (kNm/rad) | 1360 | 1406 | 3.27 |

Θ_{z} (kNm/rad) | 86.3 | 83.1 | 3.85 |

Newly Developed Stage | Conventional Stage [20] | |
---|---|---|

Settling time (2 µm step, 1%) | 48 ms | 73 ms |

Speed ripple (100 mm/s) | 0.127 mm/s | 0.3 mm/s |

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**MDPI and ACS Style**

Lee, H.-J.; Ahn, D.
Development of Air Bearing Stage Using Flexure for Yaw Motion Compensation. *Actuators* **2022**, *11*, 100.
https://doi.org/10.3390/act11040100

**AMA Style**

Lee H-J, Ahn D.
Development of Air Bearing Stage Using Flexure for Yaw Motion Compensation. *Actuators*. 2022; 11(4):100.
https://doi.org/10.3390/act11040100

**Chicago/Turabian Style**

Lee, Hak-Jun, and Dahoon Ahn.
2022. "Development of Air Bearing Stage Using Flexure for Yaw Motion Compensation" *Actuators* 11, no. 4: 100.
https://doi.org/10.3390/act11040100