# NPS6D200--A Long Range Nanopositioning Stage with 6D Closed Loop Control
## 3 Setup and Subsystems of the 6D Nanopositioning System
### Control System  
The standard structure of the control system is shown in Figure 6. The core of the control hardware is a dSpace Scalexio rapid control prototyping system [41] where the control algorithm runs as a real-time task with 10 kHz sample frequency. The controller task collects the 6D feedback signals via analog and digital inputs and calculates new target values for the different actuators based on the developed control law. The command values are the current and pneumatic pressure setpoints for the previously described planar and vertical actuators. An intermediate layer of control hardware, namely linear analog current amplifiers and high-precision pressure controllers, converts these command values into actual physical quantities--coil currents and pneumatic pressures--that are applied to the actuators in the motion stage. Figure 5: Lifting and actuation module LAU25. **1—**Air bushing as vertical guiding. **2—**Pressure chamber of pneumatic weight force compensation. **3—**Moving part (mounted to the slider). **4—**Encoder measurement system. 5—**Voice coil actuator, iron core. **6—**Voice coil actuator, coil part (moving). **7—**planar air bearing. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Subsystem** & **Parameter** & **Value** \\ \hline \multirow{3}{*}{mechanics} & \(z\)—stroke & 25 mm \\  & mass moving part & 0.7 kg \\  & mass stationary part & 1.7 kg \\  & size (\(\diameter{g}\)D\(\times\)H) & \(\diameter{g}83\) mm \(\times\) 90 mm \\ \hline \multirow{3}{*}{voice coil actuator} & force constant & 9.9 N A\({}^{-1}\) \\  & motor constant & 14.0 N/\(\sqrt{\text{W}}\) \\  & maximum current & 1.3 A \\  & current increment (LSB) & 0.04 mA \\ \hline \multirow{3}{*}{pneumatic actuator} & force constant & 80.4 N bar\({}^{-1}\) \\  & maximum pressure & 2.0 bar \\  & pressure increment (LSB) & 0.06 mbar \\ \hline \multirow{3}{*}{aerostatic guiding} & supply pressure & 4.0 bar \\  & supply air flow & 1.9 NLPM \\  & drain air flow & 1.7 NLPM \\ \hline \multirow{2}{*}{encoder feedback system} & scale pitch & 256 μm \\  & measurement resolution & 4 μm \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Key parameters of the lifting module LAU25. Figure 7 illustrates an individual axis's closed-loop block diagram. We developed six control laws for the motion controller, one for each axis, based on the classical PID approach, which is commonly used for this class of nanopositioning systems [42,43]. A set of filters then extends the aforementioned approach to attenuate disturbing resonances in the high-frequency band and, consequently, reduce positioning errors. The standard single-input single-output PID controller with dirty derivative in time-domain reads  
\[ \begin{split} v_{\text{d}}(t)&=-\omega_{\text{d}} \int_{0}^{t}v_{\text{d}}(\tau)\,\text{d}\tau+\omega_{\text{d}}\,y_{e}(t),\\ u_{e}(t)&=K_{\text{P}}\bigg{(}y_{e}(t)+K_{\text{I}} \int_{0}^{t}y_{e}(\tau)\,\text{d}\tau+K_{\text{D}}\,v_{\text{d}}(t)\bigg{)}, \end{split}  \tag{3}\]  
or equivalently in the Laplace domain  
\[G_{\text{c}}(s)=\frac{u_{e}(s)}{y_{e}(s)}=K_{\text{P}}\bigg{(}1+\frac{K_{\text{ I}}}{s}+K_{\text{D}}\bigg{(}\frac{\omega_{\text{d}}s}{s+\omega_{\text{d}}} \bigg{)}\bigg{)}, \tag{4}\]  
constitutes a proper (implementable) transfer function where \(K_{\text{P}}\), \(K_{\text{I}}\), \(K_{\text{D}}\) are positive constants. The controller parameters (\(K_{\text{P}}\), \(K_{\text{I}}\), \(K_{\text{D}}\)) and cut-off frequency \(\omega_{\text{d}}\) of the dirty derivative are tuned independently for every axis featuring sufficiently robust stability, i.e., broad bandwidths, gain margins, and phase margins. Nevertheless, the performance of the proposed controller (4) is limited by the resonance modes in the high-frequency band. Thereby, each controller additionally includes a series connection of notch filters with low-pass filters to counteract these disturbing resonances. The band-stop and cut-off frequencies are adequately selected to attenuate specific resonances in the high-frequency band.