# A Real-Time Numerical Decoupling Method for Multi-DoF Magnetic Levitation Rotary Table
## 1 Introduction  
A magnetically levitated (maglev) actuator is a promising substitute for a traditional electrical machine in modern industrial applications as it can provide accurate multiple degrees of freedom (DoF) movement benefiting from the characteristics of nonfriction and noncontact. This feature also decreases the vibration on the moving parts of the motion system because the transfer function between the displacements of the foundation and the levitation part has a low pass feature. Furthermore, the magnetic levitation system only contains a single moving part, so it possesses a compact design structure and wide control bandwidth. In the literature, taking advantage of maglev technology, there are the following five application scenarios related to intelligent instrumentation for modern industries:  
1. High-precision planar motion control to hold a wafer for a lithography machine [1; 2]. 2. Providing multiaxis motion with resolution down to nanometers and microradian in less than a millimeter for nanoscale profiling and nanoindentation [3; 4]. 3. Producing accurate force and torque in a large-volume region and unlimited rotation for haptic feedback systems [5; 6]. 4. Used as actuator for magnetic manipulation for navigation or teleoperation systems in medical robots [7; 8]. 5. Working as an active gravity compensator to reduce vibration for the stage [9; 10]. Some researchers have explored maglev technology for the computerized numerical control (CNC) micromachine system, which always requires high-precision multiaxis motion in a certain working stroke [11]. The accuracy of the positioning system in these processing devices decides the manufacturing outcomes. In a traditional design, multi-DoF motion is realized via a rotary table assembled on a three-dimensional (3D) translational stage. However, in micromilling on the mesoscale, the resolution of the positioning system should be on the micro and microradian level in a range of several millimeters, and it is hard for ordinary translational equipment and rotary tables to meet these demands. Considering the existing positioning stage based on shape memory alloys (SMAs) and piezoelectric actuators [12], a magnetically levitated actuator is an alternative. Besides the advantage of accurate positioning ability, an electromagnetic actuator has a larger working stroke and a more linear performance compared with other micromotion devices. Furthermore, using the maglev actuator, micromachining CNC devices can miniaturize as there is only single moving part in the motion system. Therefore, a magnetically levitated rotary actuator with multi-DoF translational ability in a relative large stroke is studied in this work. In general, the topology structure of maglev actuators is varied due to the real applications [13; 14; 15; 16]. In an accurate motion-control area, a reasonable structure can not only decrease the force and torque ripples but also reduce difficulty in motion decoupling. The topology of the maglev rotary actuator can refer to the magnetically levitated planar motor [17; 18] or employ the axisymmetric structure in which the magnet and coil are distributed along the circumference [11]. Once a specific design structure is given, the next step is obtaining the magnetic force and torque model [19; 20; 21; 22; 23; 24; 25]. It is also called the wrench model and is the foundation of the motion-decoupling unit in a control system. However, it is hard to consider the rotation angle and translation position in the existing modeling method at the same time, which hinders the maglev positioning system gaining translational rotational ability in a relatively large stroke [26]. In order to meet the requirements of micromachining on a mesoscale, it is necessary to study a suitable modeling method for motion decoupling in a large stroke. Plenty of analytical force and torque modeling methods have been proposed for maglev system control. These wrench models should maintain computational accuracy and efficiency.