# NPS6D200--A Long Range Nanopositioning Stage with 6D Closed Loop Control
## 3 Setup and Subsystems of the 6D Nanopositioning System
### Planar Drive System with Aerostatic Guiding  
The integrated direct drive to control the planar motion is formed by a \(120^{\circ}\) arrangement of three linear actuators (\(A\), \(B\), \(C\)), each consisting of stationary flat coils and a magnet array fixed to the slider. This is illustrated in Figure 3. At each of the actuators, the force is generated as a Lorentz force between the coils and magnets, and the phase currents in the two coils are subject to a position-dependent commutation law to have the resulting force mainly horizontal and to minimize the parasitic vertical force components. The three forces of \(A\), \(B\), and \(C\) combine to the resulting driving forces in the \(xy\) plane (\(F_{x}\), \(F_{y}\)) and the driving torque to control \(\varphi_{z}\), respectively. According to the \(120^{\circ}\) arrangement, this transformation reads as per  
\[ \begin{pmatrix}F_{x}\\ F_{y}\\ M_{\varphi_{z}}\end{pmatrix} = \begin{pmatrix}1&-\frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ r_{F}&r_{F}&r_{F}\end{pmatrix}\begin{pmatrix}F_{A}\\ F_{B}\\ F_{C}\end{pmatrix} . \tag{2}\]  
Table 3 lists the actuator constants of the individual drive units and the resulting planar motor, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline displacement measurement resolution V, W, Z & 20 pm \\ resulting feedback resolution \(x\) & 17 pm \\ resulting feedback resolution \(y\) & 10 pm \\ angular resolution \(\varphi_{z}\) & 1.4 nrad \\ angular resolution \(\varphi_{x}\), \(\varphi_{y}\) & 40 nrad \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measurement resolutions of the feedback system. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline linear actuator force constant & 6.0 N A\({}^{-1}\) \\ linear actuator motor constant & 3.8 N/\(\sqrt{\mathrm{W}}\) \\ planar drive motor constant & 4.6 N/\(\sqrt{\mathrm{W}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Key actuation parameters of the integrated planar direct drive. The slider's planar motion is guided by three planar air-bearing pads, each at the bottom of a lifting module. The reflectors are bonded directly to the slider and both are made of quartz glass to minimize susceptibility to thermal disturbances. For the same reason, the drive coils are encapsulated in a stainless steel housing with flat tempering channels on top and beneath the coils. Hence, this coil housing allows precise control of the temperature of the coil assemblies via a water cycle and a thermostat. Table 3 contains the main characteristics of the planar drive in the nominal state at the bottom of the 25 mm vertical working range. When the slider is lifted, the air gap between the coils and magnets increases beyond the nominal state, and the conditions for generating the planar drive forces change. The main effect here is a decrease in the efficiency of force generation, expressed by a decrease in the force constant of the actuators. Based on prior FEM simulations, a drop of approximately 50% was expected for a vertical displacement up to 10 mm. In view of this and with the desired vertical displacement of up to 25 mm, a coil lift functionality was developed and integrated into the stage design. The coil lift system displaces the star-shaped common base of the drive coil assemblies according to the slider's \(z\) position so that the air gap between the coils and magnets remains close to the nominal level and the conditions for the generation of the planar driving forces do not change significantly with the sliders elevation. Thereby, the tracking motion of the coil lift system does not have to be highly precise because the air gap tolerates minor height variations. The primary design challenge here is ensure adequate lateral stiffness to accomodate the reaction forces at the coils. This is solved by means of an aerostatic guiding for the moving part which was developed together with AeroLas GmbH.