## 3 2D Positioning Control
### 2D Control Strategy  
The four motors are placed in parallel pairs. Thus, as long as the motor pairs displace the same distance, the moving platform remains aligned. It was experimentally verified that the moving platform can be positioned along its working range by controlling each of the motors individually with a 1D positioning strategy (presented in [9]) implemented in each motor. Nevertheless, independently controlling each motor results in undesired rotations around the Z-axis during the transient response--that is, when the platform is moving from one position to other. Undesired rotations cause misalignments between the laser beams and plane mirrors, which can affect the performance of the laser system and even impede the displacement measurements. Therefore, in order to prevent undesired rotations of the moving platform during motion, it is necessary to coordinate the control of the four motors in a 2D control strategy. On the other hand, in [7], control of the out-of-plane motion was proven to be unnecessary due to the high stiffness of the air bearings. Similarly, in the NanoPla, the moving platform is levitated by three vacuum-preloaded air bearings with an input pressure of 0.41 MPa and an input vacuum of 15 mmHg, as recommended by the manufacturer. At these working conditions, the air bearings have a stiffness of 13 N/\(\upmu\)m. As calculated in [9], the variation of the vertical force during motion has a maximum increment of 7%, which results in negligible vibrations of the moving platform. Therefore, control of the out-of-plane motion in the NanoPla is unnecessary, although it will be monitored by three capacitive sensors. The proposed control strategy positions the platform in the X- and Y-axes and minimises rotations around the Z-axis (\(\theta_{\rm zref}\) = 0), to prevent laser system misalignments. This is done with three independent proportional-integral-derivative (PID) controllers that act on the forces generated in the X- and Y-axes by the motor pairs (\(\rm F_{x}\) and \(\rm F_{y}\)) and on the torque (\(\rm T_{z}\)) generated by the four motors. The positioning feedback (\(\rm X_{s}\), \(\rm Y_{s}\), and \(\rm\theta_{s}\)) is provided by the three laser beams (Laser Y1, Y2, and X) of the laser system. Considering the symmetry of the moving platform, the total forces and torque are divided between the four motors, and the horizontal force that each of the motors needs to generate is computed. Figure 5 illustrates the scheme of the control system that has been implemented in this project. The input of the control system is the target position (\(\rm X_{ref}\), \(\rm Y_{ref}\)) of the moving platform that is entered in the graphic user interface. In addition, the rotation around the Z-axis should be kept minimal (\(\theta_{\rm zref}\) = 0). The previously described control strategy is computed in Simulink(r) on the host PC (Figure 5). Firstly, this strategy calculates the horizontal forces that each of the linear motors needs to generate to correct positioning errors. Then, the phase currents that each linear motor requires to generate those forces are calculated according to the commutation law defined in [9]. The outputs of the control strategy are the corresponding phase voltages that the control hardware must generate for each motor. These phase voltages are generated at the power stage of each DMC kit. Then, the interaction of the phase currents flowing through the stator coils with the magnetic field of the magnet arrays of the moving platform generates horizontal forces that move the platform. This movement is recorded by the laser system and fed back to the control strategy. In order to obtain the transfer function for the rotation around the Z-axis, the four motors are set to displace from an initial rotated position to a stable equilibrium position \(\theta_{\mathrm{z,eq}}=0\). At this initial position, the angular deviation is not null.