# Air-Based Contactless Wafer Precision Positioning System
## 3 Flowerbed  
A demonstrator was built based on the bendable shaft concept. However, instead of bendable shafts, two membranes are used which allows the shafts to tilt. This reduces the force requirement of the actuator. The concept is shown in Figure 3. The top membrane connects the flowers to the world. The bottom membrane is attached to the movable plate, which is connected to an actuator. The top membrane also functions as a seal for the upper chamber, which is connected to the vacuum pressure outlet. The inlet pressure comes in through the hollow centres of the shafts.  
Figure 4 shows a cut-through view of the Flowerbed. The Flowerbed has 61 hexagonal heads, with an in-radius of 14 mm. It is nicknamed the Flowerbed because the surface looks like flowers that tilt towards the sun. The hexagon shape of the flowers was selected because it provides a full packing of the surface in combination with an optimal circumference to surface area ratio, and thus a maximum force for a given airflow. Furthermore, the figure also shows the fixed plate and movable plate, where both consist of two plates that are clamped together with a 50 um thick membrane in between. The membrane is also clamped in the flowers. To better align the flowers, the movable plate is preloaded downwards with three springs, with a force of approximately 100 N. After assembly, the final step was the spark erosion of the surface with wire-cut EDM technology to further level the surface.  
Figure 3: Schematic of the second generation contactless actuator, showing the flow of air through the system, for a displacement \(d\) of the movable plate [13].  
Figure 2: The tilting surface concept, the surfaces are titled by bending the shafts [4].  
Figure 5 shows a single flower in both a neutral and actuated position. The thin film airflow is visualized in the area between the effective surface of the flower and the substrate. The combination of the positive and negative pressures provides stiffness in the out-of-plane direction at the fly-height \(h\).  
The flowers are tilted with an angle \(\alpha\), through a displacement of the movable plate, as shown in Equation (1). Where \(d_{p}\) is the displacement of the movable plate and L is the distance between the movable plate and the fixed plate.  
\[sin(\alpha)=\frac{d_{p}}{L}\approx\alpha \tag{1}\]  
when tilted the majority of the airflow will follow the path of least resistance. This flow will, in turn, exert a viscous traction force \(F_{w}\) on the wafer in addition to the out of plane stiffness.  
Figure 4: A cross-section of the Flowerbed showing the fixed plate, flowers, and movable plate. Source: [13].  
Figure 5: Schematic of a single flower of the Flowerbed in (**a**) neutral position and (**b**) actuated position. The force on the wafer is visualized in the actuated position [13].  
The tilting angle of the flower has been measured to be proportional to the displacement of the movable plate up to 1000 Hz, without delay. The tilt response between the individual flowers is similar, there is only a small deviation in the gain. The values of gain vary between 71 rad/m and 80 rad/m, where the predicted value is 77 rad/m [4]. Furthermore, [PERSON] experimentally validated that the pneumatics can be modelled by a proportional gain, up to at least 400 Hz. This means that if the movable plate position is known, the force exerted on the wafer is known.  
The stiffness of the Flowerbed is different for the translation and rotation of the movable plate. Each flower has an elastic hinge at the top and bottom, which provide stiffness against rotation. Figure 6, shows the behaviour of the Flowerbed during the translation and rotation of the movable plate. The stiffness and inertia values for translation and rotation result in two suspension eigenvalues. These values have been calculated and experimentally validated by [PERSON] [13]. The important properties are summarized in Table 1.