## 4 Investigations and Measurement Results
Furthermore, the angular deviation during a line movement with a length of 100 mm at the maximum speed was considered. During line travel in the \(x\) direction, an angle deviation (1\(\sigma\)) of 714 nrad was measured. In the \(y\) direction, this measured deviation is 842 nrad. In Figure 10, five consecutive revolutions with a nominal trajectory with a radius of 1 mm are shown. The movements were carried out at a circumferential velocity of 10 mm/s. In order to illustrate the deviations of the circular path more clearly, the original measurement data were transformed into polar coordinates, which is depicted in the following Figure 11.  
Figure 8: Positioning noise (1\(\sigma\)) of the x and y interferometer in \(x\) and \(y\) direction from position \(-50\) mm to \(50\) mm in a distance of 2 mm. This noise was recorded for a time of 10 s each with a frequency of 10 kHz. Figure 10: Positioning accuracy of a circle movement by the NFM-100 with a radius of 1 mm and a circumferential velocity of 10 mm/s. Here, the standard deviation of the radius is 3.107 nm (1\(\sigma\)). The angular \(\varphi_{Z}\) deviation in this case is 354 nrad. Figure 9: Straightness deviation for movements in \(x\) and \(y\) direction with a length of 50 mm and 100 mm. The dependence of the deviation of the line movement was investigated in a velocity range between 0.5 mm/s and the maximum velocity of 20 mm/s. In addition to this investigation, Figure 12 shows the path deviations of a circular movement as a function of velocity. Different radii were driven with the machine table at various velocities. The examined velocity range was between 0.1 mm/s and 20 mm/s. In this examination, circles with radii between 1 mm and 40 mm were driven at different speeds with five repetitions each. Also in this case, the standard deviation for the radius at the maximum velocity of 20 mm/s at r = 40 mm is 7.269 nm. For a circular movement with a diameter of 80 mm and a velocity of 20 mm/s, the maximum angular deviation is only 689 nrad. As mentioned in Section 3, the selected air bearings are characterised by numerous advantages. However, a residual noise in the \(z\) direction is still present. To investigate this, the existing atomic force microscope (AFM) was used. An AFM scan was made on an area of 10 um \(\times\) 10 um on a sample when the machine table was switched off and, thus, the machine table was directly resting on the granite machine base without an air layer in between. For comparison, a scan was measured when the table was switched on. For this purpose, a sample with a surface that has only very small steps (<1 nm) in the \(z\) direction was selected. Here, a SiC sample was used that has step heights of about 750 pm and step widths between 150 nm and 400 nm. The step height is equal to half of the lattice constant of the 6H-SiC crystal in the [0001] direction [28]. As shown in Figure 8, the position noise in the \(x\) and \(y\) direction is below 300 pm, which can be neglected for the AFM scan in that case. Figure 13 (left and right) shows a topography scan of the aforementioned SiC sample with the difference between the machine table being switched off and on. The scanning speed for both measurements was 5 um/s. The individual steps are clearly visible in both pictures. However, in the picture with the machine table switched on, it can be seen that the amount of noise increases slightly. In addition, a section of the two topography profiles was plotted in Figure 14 for comparison. The root mean square roughness is only 90 pm for the measurement with  
Figure 11: Radius deviation of a circle movement with 1 revolution (\(\theta=2\pi\), unwinding of circle in Figure 10). Figure 12: Deviation of circle movement of a given circle trajectory with radii between 1 mm and 40 mm at different velocities (0.1 mm/s–20 mm/s). The circular movement was carried out with 5 revolutions in each case. the machine table switched off. In contrast, a roughness with the table switched on in closed-loop control was 144 pm.