## 3 Setup of the Planar Nanopositioning Machine
The two interferometers for length measurement are arranged in such a way that their measuring axes virtually intersect at the probing point of the measuring tool, so that the Abbe principle can be fulfilled in the \(x\), \(y\) and \(z\) direction with an Abbe-offset \(l_{Abbe,i}=0\)[17] and an uncertainty \(u_{Abbe,i}<100\,\mathrm{\SIUnitSymbolMicro m}\):  
\[l_{Abbe,i}=0\pm u_{Abbe,i} i\in x,y,z \tag{1}\]  
This way, first-order rotation-induced length measuring errors can be reduced to a minimum. Otherwise, the tilt \(\varphi\) of the slider around the \(x\), \(y\) or \(z\) axis would lead to an Abbe-error \(\Delta l\)[17]:  
\[\Delta l_{i,j}=u_{Abbe,i}\cdot sin(\varphi_{j})\approx 0 i,j\in x,y,z. \tag{2}\]  
The third interferometer, which is used for angle measurement, is arranged in such a way that the reference beam and the measuring beam both hit the moving mirror mounted on the machine slider. Based on the known spacing and the length difference of both interferometer beams, the rotation of the machine table around the z axis can be measured and compensated for by the control system. This is especially necessary as the mechanical design of the NFM-100 does not constrain this rotation. Therefore, the rotation must be actively measured and controlled to zero. The maximum rotation allowed here is \(30\,\mathrm{\SIUnitSymbolArcArc}\). Because the rotor has no mechanical rotation lock, closed-loop operation is necessary. A dSpace(r) system is used here as the control system. The following table (Table 1) shows the technical specifications of the nanofabrication machine presented here [26]. In Figure 4, the basic structure of the control system is depicted. The measured variables are the three length values of the laser interferometers (\(U,V,W\)). These are transformed to the positioning coordinates (\(x,y,\varphi_{z}\)). The rotation of the stage around \begin{table}
\begin{tabular}{c c} \hline \hline Positioning range & \(\mathcal{O}100\,\mathrm{mm}\) \\ Maximum velocity & \(20\,\frac{\mathrm{mm}}{\mathrm{s}^{2}}\) \\ Maximum acceleration & \(250\,\frac{\mathrm{mm}}{\mathrm{s}^{2}}\) \\ Interferometer resolution & \(5\,\mathrm{\SIUnitSymbolMicro m}\) \\ Maximum added mass & \(2\,\mathrm{kg}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the positioning machine NFM-100. Figure 3: CAD model of the Nanofabrication Machine (NFM-100). The AFM measuring system is mounted directly above the centre of the machine table. the \(z\)-axis \(\varphi_{z}\) is calculated from towards the Abbe point symmetrically arranged probe beams of the \(U\) interferometer. \(\varphi_{z}\) is derived from this separation per angular displacement \(\Delta l\approx 14\,\mathrm{mm/rad}\). The angular displacement \(\Delta_{\varphi}\leq 50\,\mathrm{arcsec}\) are kept very small which allows for linearisations of trigonometric functions. Furthermore, the apparent cosine error originating from the distance between origin and \(U\)-mirror (compare Figure 2) becomes negligible. The implemented simplified reconstruction of the \(x\)- and \(y\)-axis from the measured length values states as:  
\[T_{VWUlxy\varphi}=\left( \begin{array}{ccc}-\frac{\sin(\alpha_{W})}{\sin( \alpha_{V}-\alpha_{W})}&\frac{\sin(\alpha_{V})}{\sin(\alpha_{V}-\alpha_{W})}&0 \\ \frac{\cos(\alpha_{W})}{\sin(\alpha_{V}-\alpha_{W})}&-\frac{\cos(\alpha_{V})}{ \sin(\alpha_{V}-\alpha_{W})}&0\\ 0&0&\frac{1}{\Delta l}\end{array} \right) \tag{3}\]  
where \(\alpha_{W}=-30^{\circ}\) and \(\alpha_{V}=-150^{\circ}\) are the rotation angles of the \(V\) and \(W\) axis starting from \(x\) axis. The core of the control system is formed by three separate classic PID controllers for each of these coordinates [24]. For \((x,y,\varphi_{z})^{T}\), an independent PID controller in its standard form is employed:  
\[\vec{a}(t)=K_{P}\odot\vec{e}(t)+K_{I}\odot\int_{0}^{t}\vec{e}(\tau)d\tau+K_{D }\odot\frac{d\vec{e}(t)}{dt} \tag{4}\]  
where \(\odot\) denotes the (element-wise) Hadamard product.