## 3 2D Positioning Control
### Dynamic Characterisation of the System  
In the NanoPla, the motors are placed in parallel pairs (Figure 3); thus, two motors, motor 1 and motor 2 (represented as M1 and M2 in Figure 3b), generate forces on the X-axis (\(\mathrm{F_{M1}}\) and \(\mathrm{F_{M2}}\)) that move the platform in the X-direction. Similarly, the other two parallel motors, motor 3 and 4 (M3 and M4), generate forces on the Y-axis (\(\mathrm{F_{M3}}\) and \(\mathrm{F_{M4}}\)) that move the platform in the Y-direction. In addition, the four motors are placed symmetrically at a distance R (169.9 mm) from the centre of the platform and, thus, their forces generate a torque at the centre of the moving platform, around the Z-axis. The movements of the platform in the X- and Y-axes -\(\mathrm{X_{s}}\) and \(\mathrm{Y_{s}}\)- and the rotation around Z-axis -\(\mathrm{\theta_{zS}}\)- are monitored by the 2D laser interferometer system (Laser Y1, Y2, and X). The total forces in the X- and Y-axes and the torque around the Z-axis -\(\mathrm{T_{z}}\)- can be calculated as follows:  
\[\mathrm{F_{x}}\ =\ \mathrm{F_{M1}}+\mathrm{F_{M2}} \tag{1}\]  
\[\mathrm{F_{y}}\ =\ \mathrm{F_{M3}}+\mathrm{F_{M4}} \tag{2}\]  
\[\mathrm{T_{z}}\ =\ -\mathrm{F_{M1}}\cdot\mathrm{R}+\mathrm{F_{M2}}\cdot\mathrm{R}- \mathrm{F_{M3}}\cdot\mathrm{R}+\mathrm{F_{M4}}\cdot\mathrm{R} \tag{3}\]  
The dynamic model of a Halbach linear motor working as a positioning actuator in a pneumatically levitated linear stage was identified as a servosystem in a previous work [14]. In this system, the electromagnetic horizontal force generated by the motor around the stable equilibrium position (\(\mathrm{F_{x}}\) = 0, \(\mathrm{F_{z}}\) > 0) acts as a proportional controller. The closed-loop transfer function that relates the equilibrium position, \(\mathrm{X_{eq}(s)}\), to the position of the stage, \(\mathrm{X(s)}\), was identified with a spring-mass-damper model (second order system). The NanoPla 2D positioning model is expected to present a similar second-order transfer function in each axis of motion, since it uses the same actuators and the moving platform is also pneumatically levitated. The transfer function of the system can be obtained experimentally, which enables a better understanding of the system and allows tuning of the controllers in advance, which facilitates tasks in the experimental setup. The force generated by each Halbach motor around the stable equilibrium position (linear zone) can be defined as  
\[\mathrm{F_{M,x}(s)}\ =\ \mathrm{K_{M}(X_{eq}(s)-X(s))} \tag{4}\]  
where \(\mathrm{K_{M}}\) is the slope of the thrust force generated by each Halbach linear motor around the equilibrium position. The forces along the X-axis generated by the parallel motor pair M1 and M2 at the initial position are represented in Figure 4. The linear zone and the slope around the stable equilibrium position are also represented in Figure 4. As shown, the linear zone has a length of approximately 5 mm, with the stable equilibrium position in the middle of the region. In this system, the stable equilibrium  
Figure 3: Scheme of the forces that act on the moving platform: (**a**) isometric view; (**b**) top view scheme. position of the parallel pair of Halbach linear motors M1 and M2 is set at the same X-coordinate. Thus, the total force, F\({}_{\times}\), generated around the stable equilibrium position is twice the force defined in Equation (4). Therefore, the transfer function that relates the final position of the stage X(s) with the defined stable equilibrium position Xeq(s) is the following:  
\[\frac{\mathrm{X(s)}}{\mathrm{Xeq(s)}}\ =\ \frac{\frac{2\mathrm{K_{M}}}{ \mathrm{m}}}{\mathrm{s^{2}+\frac{\mathrm{b_{X}}}{\mathrm{m}}\mathrm{s}+\frac{ 2\mathrm{K_{M}}}{\mathrm{m}}}} \tag{5}\]  
where \(m\) is the mass of the moving part and \(b_{X}\) defines the viscous-friction elements of the setup and the eddy-current damping of the motors in the X-axis.