## 5 Conclusions  
In this work, a positioning control system for a 2D nanopositioning stage was designed and implemented in the NanoPla. The proposed control system drives four Halbach linear motors that allow planar motion, while a 2D plane mirror laser interferometer system works as the positioning sensor. The selected control hardware is a Digital Motor Control kit from Texas Instruments for generic three-phase motors. The target is to obtain an accurate positioning control system that fulfils the NanoPla requirements by implementing commercial hardware without any additional electronics.  
The NanoPla offers a two-stage scheme that complements the XY-long range positioning of the moving platform (50 mm \(\times\) 50 mm) with an additional commercial piezo-nanopositioning stage that is fixed to the inferior base. This second stage works in a range of 100 \(\upmu\)m \(\times\) 100 \(\upmu\)m. Due to this, the maximum positioning error requirement of the NanoPla in the X- and Y-axes has been decided to be 10 \(\upmu\)m. In addition, the rotation around the Z-axis must be kept minimal in order to avoid the laser system's misalignment. In this article, a dynamic model for a NanoPla with four Halbach linear motors as actuators was first identified. Then, a control strategy for the positions in the X- and Y-axes and the rotation around Z-axis was designed and implemented for the control hardware. The correct performance of the proposed control system has been experimentally verified for the NanoPla. In addition, the positioning uncertainty of the control system has been computed, and its contributors analysed. The obtained positioning uncertainty U\({}_{X}\) = U\({}_{Y}\) = U\({}_{XY}\) (\(k=2\)) is equal to \(\pm\)0.50 \(\upmu\)m in each axis and in the entire working range of the NanoPla. Therefore, the resulting positioning uncertainty of the control system implemented in commercial generic hardware without additional electronics is much lower than the initial NanoPla requirements, thus broadening the applicability of the designed positioning system.  
In future work, the control strategy could be improved to minimise the coupling between axes during movement. In addition, possible alternatives to improve global uncertainty with additional electronics will be studied. Future research should also focus on the implementation of a measuring system in the NanoPla. As previously noted, the target application of this first prototype is surface topography characterisation at the atomic scale of samples with relatively large planar areas, using an AFM. Nevertheless, due to the fragile configuration of the AFM system, the implementation of a confocal sensor as an intermediate solution before integrating the AFM is proposed.  
[PERSON], [PERSON], [PERSON], and [PERSON] developed the positioning control system of the NanoPla.; [PERSON] performed the experiments and wrote the original draft of the manuscript. [PERSON], [PERSON], and [PERSON] reviewed and edited the manuscript.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Source** & **Justification** & **Standard Uncertainty** & **Relative Contribution** \\ \hline Resolution at the high-resolution & Resolution of 26.2 \(\upmu\)V & 0.7/12 \(\upmu\)m & 65.1\% \\ PWM (PWM) multigrid & Resolution of 1.58 nm & 1.58/\(\upmu\)12 nm & 0.00\% \\ Laser system resolution u\({}_{\text{L}\text{o}\text{i}\text{o}\text{i}\text{o}\text{i}\text{o}\text{i}\text{o}\text{i}\text{o}\text{i}\text{o}\text{i}\text{o}\text{o}\text{i}\text{o}\  
**Funding:** Funded by the Gobierno de Aragon (Reference Group T56_17R) and co-funded with Feder 2014-2020 \"Construyendo Europa desde Aragon\" and the Spanish government project RTI2018-097191-B-I00 \"MultiMet\" with the collaboration of the Diputacion General de Aragon--Fondo Social Europeo.  
**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.