## 3 2D Positioning Control
### 2D Control Strategy
In order to define the initial position, the four motors are displaced at a distance of \(\Delta\theta_{\mathrm{z}}\cdot\)R, equal to 15 \(\upmu\)m. In each motor pair, each motor displaces in an opposite direction to contribute to the rotation around the central point of the stage, generating an angular deviation of \(8.83\times 10^{-5}\) rad, which is close to the maximum displacement allowed without losing the laser system's alignment. The experimentally obtained values for \(\mathrm{I_{z}}\) and \(\mathrm{K_{M}}\) approximately match the actual inertia of the moving platform and the slope of the thrust force around the stable equilibrium position, respectively. Nevertheless, due to the limited range of the angular deviation and the short response time, the recorded response is not smooth enough to perfectly match the simulated plant. Figure 7 represents the experimentally obtained \(8.83\times 10^{-5}\) rad step response of the stage around the Z-axis (\(\theta_{\mathrm{s}}\)). The simulated response of the adjusted model is also shown (\(\theta_{\mathrm{sim}}\)). After the transfer function identification, the proposed control system was implemented in the NanoPla, and its correct performance was experimentally verified. For these experiments, the vertical force generated by each motor was defined as 2 N, which limits the phase currents' working range to \(\pm 0.83\) A.  
Firstly, the stability of the positioning control system was examined for a long period of time. The moving platform was set to remain still at the initial position for 30 min, and it was experimentally verified that the position deviations are confined to \(\pm 1\)\(\upmu\)m. In Figure 8, a 400 s period of this test is shown, and the root mean square (RMS) deviation during this time is 0.28 \(\upmu\)m. The results in the Y-axis are similar, since the moving platform is symmetrical. During this period of time, the force generated by each linear motor varied between \(\pm 2\) mN--that is, the system worked around the stable equilibrium position, as expected. Figure 7: Angular step response of \(8.83\times 10^{-5}\) rad around the Z-axis of the stage in the open loop and simulation of the plant. As mentioned earlier, the NanoPla has a two-stage scheme. The moving platform performs coarse movement in a large working range of 50 mm \(\times\) 50 mm. Once the moving platform arrives at the target position, it stays static (air bearings off), and a piezo-nanopositioning stage placed on the inferior base performs the fine displacement required for the scanning task. The selected commercial piezostage has a working range of 100 \(\upmu\)m \(\times\) 100 \(\upmu\)m in the XY-plane. Therefore, as mentioned, it is defined as a requirement that the position control system has a positioning error smaller than 10 \(\upmu\)m, so this error can be corrected by the fine motion of the piezo stage. Therefore, the performance of the control system was tested when performing a displacement to the target position. Then, 10 \(\upmu\)m step responses were taken in the X- and Y-directions. At the same time, the perturbation to the other axis was also recorded, as shown in Figure 9. The perturbed motions in the other axis demonstrate that there is a dynamic coupling between axes. This is unavoidable because there is only one moving part that is affected by the vibrations of the four motors. This perturbation generates a displacement on the Y-axis of a maximum of 2 \(\upmu\)m during the transient response. Nevertheless, once the NanoPla achieves its target position on the X-axis, the positioning error on the Y-axis is corrected. In addition, it has been observed that during the transient response, the maximum angular deviation is \(1.5\times 10^{-5}\) rad, which is inside the tolerance of \(\pm 1.2\times 10^{-4}\) rad required for the laser system to read. Figure 8: Long-term stability analysis of the developed positioning control system. Figure 9: The 10 \(\upmu\)m step response on the X-axis (**a**) and perturbation on the Y-axis (**b**).