## 1 Introduction  
In modern high-end manufacturing or testing equipment, such as a wafer stage in a lithography machine, micro/nano positioning platform, and scanning probe microscope, the requirements of the high-precision multi-degree-of-freedom actuating elements are increasing [1; 2; 3; 4; 5; 6]. Generally, there are three conventional methods to realize multi-degree-of-freedom positioning motions. The first method is the flexure hinge mechanism driven by piezoelectric ceramics [7; 8; 9]. The second method is to use multiple rotary motors and the corresponding transmission mechanism. The third method is to use multiple linear motors [10; 11]. Although the multi-degree-of-freedom motions can be realized using these methods, there are some inevitable defects. For the flexure hinge mechanism and piezoelectric ceramics, it is easy to achieve integral structure and low power loss. However, the damping, micron-grade stroke of the flexure hinge and the hysteresis, non-linear characteristics of the piezoelectric ceramics will limit the performance improvement. For the rotary motor systems, the friction, backlash, and deformation of the transmission mechanism will degrade the positioning accuracy. Additionally, the additional mass of the transmission mechanism is a bottleneck to improve the dynamic performance. For linear motor systems, the positioning accuracy and the dynamic performance is improved due to removal of the transmission chain. However, it still cannot get rid of the mode of superposing low dimensional motions into a high dimensional motion. The bottom motor has to support the top motor. Additionally, a large number of mechanical components leads to a complex structure, which is adverse to further improving the positioning accuracy. In the last two decades, planar motors brought about a novel and practical solution for industry, and they have been continually receiving more and more attention. A planar motor is a type of motor that can realize planar motion with only one moving part. Planar motors have the features of two-degree-of-freedom (2-D) direct drive, high response, high structural strength, and simple system structure. Therefore, it has broad application prospects in 2-D positioning systems, especially in the high precision applications. According to the working principle, planar motors can be divided into four types, which are reluctance-type planar motor, induction-type planar motor, synchronous-type planar motor, and Lorentz-force-driven planar motor. A planar stepping motor belongs to reluctance-type motors and it is the earliest planar motor [12; 13]. A planar stepping motor is comprised of two linear stepping motors. The movers of the two linear stepping motors are integrated into one moving part in mutually perpendicular directions. The stator is a 2-D tooth-slot array. The advantages of the planar stepping motor are no accumulated error and simple structure, but the low frequency oscillation, losing steps, and low speed limit its range of applications. A switched reluctance planar motor is another type of reluctance-type motor, which was proposed by [PERSON] et al. [14; 15]. The stator is composed of several blocks, each of which is made of silicon steel sheets instead of solid iron. Therefore, the eddy current effect and the processing cost is reduced. In 1999, [PERSON] et al. [16] proposed a planar induction motor. The primary is an iron ring winded with windings, and the secondary is a magnetic plate. This motor can achieve not only the long-stroke planar motion, but also the rotary motion. However, there are some disadvantages, such as its complex manufacturing process, low air-gap magnetic flux density, and low operating speed. The 2-D electromagnetic force of the synchronous-type planar motor is from the interaction between a permanent magnet array and coil array. In recent years, this type of planar motors was deeply researched, and many novel topologies were put forward.