## 2 Materials and Methods
### Components of the Control System  
The positioning systems' main components are actuators, control hardware, and positioning sensors. The selection of these components is based on the positioning stage's required precision, as well as the operating range and structure of the stage. Similarly, the development of the control strategy is constrained by the positioning system components defined by the NanoPla's design. In addition, the implementation of the components in the positioning control system can be optimised to leverage the capabilities of each of component in order to obtain the maximum positioning accuracy. Halbach linear motors directly transform electrical energy into linear motion and have been selected as actuators in the NanoPla because of their many advantages in precision engineering due to their lack of mechanical transmission elements (thereby avoiding, for example, backlash). Similarly, contactless unguided motion prevents friction and allows planar motion. Planar motion is preferred in precision applications because it minimises geometrical errors and presents many other advantages in precision engineering [11]. The Halbach linear motors used in the NanoPla were developed by [PERSON] et al. [5] and are not commercialised. Therefore, they have been custom-made at the University of North Carolina at Charlotte, and the size of their winding areas is large enough to allow planar movement in the 2D working range of the NanoPla. When DC current flows through the three-phase coils of a Halbach linear motor, the electric field interacts with the magnetic field of the magnet array, resulting in two orthogonal forces--one horizontal and the other vertical. The relationship between  
Figure 1: Exploded view of the nanopositioning platform (NanoPla). the phase currents and the generated forces is defined by the motor law presented in [9]. The vertical forces of the four motors facilitate the moving platform's levitation, while the horizontal forces move the platform in the XY-plane. A generic commercial control system for custom-made Halbach linear motors is not currently available. For this reason, other reviewed stages integrating these motors specifically designed and developed control hardware and software for this purpose [7; 8]. In these stages, control strategies are based on individually and independently controlling each of the phase currents of the motors with transconductance power amplifiers. However, following the NanoPla's design principle of integrating commercial devices when possible, the use of only commercial hardware and no custom-made electronics was proposed. Thus, a Digital Motor Control Kit (DMC) DRV8302-HC-C2-KIT from Texas Instruments was selected as the control hardware. The DMC kit consists of a F28035 control card and a DRV8302 board. This DMC kit has been designed for rotary brushless DC and permanent magnet synchronous three-phase motors, where the aim is to control the rotation speed or the torque generated. The board includes a three-phase power stage that drives and generates phase voltages by pulse-width modulation (PWM), in contrast to other works [7; 8] where the hardware acted as a controlled current source. Additionally, the control hardware forces the star connection of the phases, thereby impeding the phase currents from being controlled independently, as was done in [7; 8]. This hardware has been selected due to its relatively low cost and the advantages of the associated software. The use of the microcontroller from Texas Instruments is based on the Target Support PackageTM for Embedded Code. This Package integrates MATLAB(r) and Simulink(r) with Texas Instrument tools and C2000 processors to generate, compile, implement, and execute the optimised control code with a user-friendly graphic interface and without programming in a specific language. As mentioned, in the NanoPla, a 2D laser system is used as a positioning sensor to provide control system feedback. This 2D laser system is a combination of three 1D plane mirror laser interferometer systems.