## 4 Positioning Uncertainty of the Control System
The actual phase  
Figure 13: Block diagram of the data flow in the control system of the NanoPla. currents have been experimentally measured using a data acquisition system (DAQ) from National Instruments. This DAQ is able to record the measurements at a frequency of 500 kHz and with a resolution of approximately 2 mA. In Figure 14, an experimentally measured phase current is compared to the simulated one, and, as shown, the current ripple in both cases is almost coincident. Nevertheless, the experimentally measured phase current includes additional noise. This deviation has different sources, including the DAQ's own noise. One of the contributors is the noise of the DC power supply that feeds the power stage of the control hardware. The noise of the DC power supply is imprinted in the PWM phase voltages and, thus, is transmitted to the phase currents. To minimise this contributor, a low-noise power supply, with a peak-to-peak noise of 10 mV, has been used. In a previous study [16], a self-calibration procedure for the geometrical characterization of the 2D laser system of the NanoPla was proposed. The standard uncertainty of the calibrated laser system, after correcting for geometrical errors, was calculated to be 99 nm on the X- and Y-axes. The laser system's resolution is 1.58 nm, and the RMS deviation of the laser readouts of an axis is 6 nm. In addition, the stability of the 2D laser system integrated in the NanoPla was also verified. In the control system of the NanoPla, the phase current noise generates deviations in the forces that act on the moving platform, thereby producing undesired vibrations in the platform. These vibrations are recorded by the laser system, adding to the laser system's own noise, and fed back to the control strategy. This results in positioning noise of the moving platform that has been experimentally measured and computed as a short term (30 s) RMS positioning error that is 0.11 \(\upmu\)m in each axis. It has also been experimentally verified that the main contributor to the RMS positioning error is the phase current's noise. NanoPla positioning uncertainty contributors can be divided into two categories: Ones whose contribution to the final positioning error in an open loop is known, like the resolution of the phase voltage generation (analysed in [14]) and the laser system (analysed in [16]); and the errors in the laser system that are still present after correcting for the geometrical errors obtained by the self-calibration procedure defined in [16]. The other types of errors are those whose contributions to the final positioning error cannot be calculated separately and cause RMS deviations of the positioning errors. Table 1 illustrates a calculation of the positioning uncertainty according to ISO/TR 230-9:2005 [17] and its contributors. Even though the laser system's resolution is included inside the standard uncertainty of the laser system, its value is also shown separately in the table, so it can be compared to the magnitude of the other contributors. Figure 14: Current ripple of the phase currents generated by the pulse-width modulation (PWM)-controlled phase voltage. The resulting positioning uncertainty U\({}_{XY}\) (\(k=2\)) in each axis and in the entire working range of 50 mm \(\times\) 50 mm is equal to \(\pm\)0.50 \(\upmu\)m. The main contributor is the HRPWM module resolution, which contributes partially to both. That is, when the laser system detects this error, the controller acts on the horizontal force to correct it, thereby resulting in oscillations. The phase currents' noise is another main contributor to the positioning uncertainty. None of these errors can be corrected without additional electronics. Nevertheless, the resulting positioning uncertainty is much lower than the initial working requirements of the NanoPla, so the developed positioning control system is considered valid.