## 4 Positioning Uncertainty of the Control System  
The control system of the NanoPla has been optimised to reduce positioning errors. The remaining positioning errors are mainly caused by the resolution of the system components and electronic devices' noise. The computing operation in the control hardware is performed with finite numbers,  
Figure 11: (**a**) The 25 mm displacement at a constant speed on the X-axis; (**b**) the 1 mm forward and backward displacement at a constant speed on the Y-axis. Figure 12: Circular motion performed simultaneously on the X- and Y-axes. which implies a rounding operation, resulting in a truncation error that, depending on its magnitude, may not be negligible. In addition, the errors due to electronic device noise cannot be completely eliminated and result in positioning noise. Thus, the positioning uncertainty of the control system is assessed and analysed in this section. In Figure 13, the control system dataflow is represented in a block diagram. The data type of the transmitted information is represented by the coloured arrows. The control strategy is computed in Simulink(r) (MATLAB(r)). Xref and Yref are the desired positions on the X- and Y-axes. The real position of the NanoPla is measured by a 2D laser system and extracted by MATLAB into the Simulink program. The control strategy computed in the PC by Simulink uses a 64-bit double-precision floating point format (blue arrows), and, in this case, the rounding operation has no significant influence on the calculated results. The control strategy outputs are the required phase voltages, contained in a range of \(\pm\)6 V, and are sent to the control hardware by a serial communication interface (SCI) (green arrows). The MCU of the control hardware works with 32-bit data types, and the voltage values are transmitted as 32-bit fixed points with a 25-bit fraction length (red arrow). The resolution derived from the data type used for the voltages values is 0.0298 nV. These phase voltages are generated in the power stage of the control hardware by PWM. The DMC kit includes a high-resolution PWM (HRPWM) module that is capable of extending the time resolution capabilities of the PWM function. Thus, the resolution of the voltage generation is defined by the time resolution of the HRPWM module and is 26.1 \(\upmu\)V. This resolution is sufficient to perform a minimum incremental motion of approximately 700 nm in an open-loop. Therefore, this hardware is not able to generate the exact combination of phase currents for every target position. Nevertheless, when working in a closed loop, the positioning controller is capable of partially correcting this error by switching between combinations of phase currents [14]. In Figure 13, the real-world values are represented with black arrows. These values are the generated phase voltages, the derived currents, and the resultant forces. In addition, the 2D laser interferometer system measures the moving platform's displacement. The uncertainty of the laser system measurement also affects the positioning uncertainty of the control system. It must also be taken into account that the PWM-controlled phase voltages lead to a ripple in the phase currents [15]. This current ripple is directly related to the inductance of the stator coils. The resistance and inductance of the coils have been experimentally measured and are 0.88 \(\Omega\) and 0.24 mH in each phase coil. The electric circuit has been simulated in order to calculate the current ripple derived from the PWM-generated phase voltages. The current ripple has a sawtooth waveform with a frequency of 29.28 kHz, which is double the PWM frequency. Moreover, the current ripple peak-to-peak value is dependent on the duty cycle (DC) of the PWM voltages; in this study, the current's working range is \(\pm\)0.83 A, which corresponds to a DC between 43.08% and 56.92%. For these values, the peak-to-peak value of the current ripple has a maximum magnitude of 0.09 A.