## 1 Introduction  
The face gear transmission is a gear mechanism that relies on the meshing of cylindrical gears and face gears to transmit motion and power between two intersecting or crossed shafts. Due to its simple support system, high contact ratio and the structure of a split-torque transmission which can reduce the weight of the gearbox and improve bearing performance, it is successfully applied in helicopter main reducers [1; 2] and in some low-speed and light load situations [3], and has broad application prospects in the future. At present, the manufacturing methods of face gears include common gear shaping, hobbing, and recently emerged power skiving [4; 5]. Fellows Company [6] was the first to propose a manufacturing method of face gears in which the gear shaping cutter is identical to the pinion, without considering the sensitivity of installation errors to the tooth surface. [PERSON] [7; 8] proposed the gear shaper cutter with tooth number difference to process face gears, limiting the contact area of the tooth surface to local areas, causing the contact method to be changed from line to point, effectively avoiding edge contact. However, this method has obvious idle stroke and discontinuous cutting process, resulting in low efficiency. [PERSON] [9] and the Crowngear corporation [10] proposed a gear hobbing method to manufacture face gears, which can improve machining efficiency through continuous indexing. However, this method has a large machining error and cannot meet the accuracy requirements. So as to improve machining accuracy and production efficiency, [PERSON] [11] put forward a method for machining face gears with worm hobs and demonstrated its effectiveness through experiments, but a new machine tool is needed. [PERSON] [12] presented a method for machining orthogonal spur gears with power skiving, and the tooth surface deviation between skiving and shaping was compared. The feasibility of this method was verified through experiments, and the influence of the parameters of the skiving cutter on meshing performance was analyzed. [PERSON] [13] analyzed the influence of parameters of skiving cutter on tooth surface deviation and proposed a deviation correction method for skiving face gears. [PERSON] [14; 15] presented a design method of skiving tool to avoid negative working clearance angles for internal gears and realized the technology on the six-axis CNC skiving and milling center. The research on the soft tooth surface of the rough machined gear mentioned above is based on the principle of gear shaping machining, which requires the cutters to imitate the theoretical tooth profile of the gear shaping cutter and make point contact with the workpiece, which limits the machining efficiency in terms of contact mode. When the hard tooth surface is high-precision machined, the gear grinding method is usually used. [PERSON] [16] proposed the method of grinding surface gears with worm grinding wheels, which improves machining efficiency and tooth surface accuracy. However, due to the fact that the worm tooth surface is an elliptical involute worm, the tooth shape is complex and difficult to the machine and dressing. [PERSON] [17] studied the method of generating machining face gears with the grinding disk and its tooth surface modification method. This method is easier to achieve than worm wheel, but the machining efficiency is lower. [PERSON] [18] presented a grinding method of face gear mating with a conical spur involute pinion. In the above research on gear grinding, the former has a higher machining efficiency than the latter. However, the disadvantage of the former is that the tool is complex and difficult to manufacture and in dressing. To simplify cutting tools and achieve low-cost machining, [PERSON] [19] proposed a new type of plunge milling cutter for machining face gears. Although it has certain universality, this tool also imitates the profile of the gear shaping cutter and performs point contact machining, resulting in low efficiency.