## 1 Introduction  
Cycloid-pin drives have the advantages of high transmission ratios, transmission accuracy, torsional stiffness, transmission efficiency, and compact structures. Cycloid-pin drives have been widely used in high-precision equipment, such as industrial robots, machine tools, and assembly and conveying equipment. This type of equipment usually requires frequent forward and reverse rotation during work and has high position accuracy requirements, which mainly depend on the transmission error (TE) and backlash of cycloid-pin drives. Establishing a calculation method for the TE and backlash, while comprehensively considering the effects of profile modifications (PMs), manufacturing errors (MEs), and assembly errors (AEs), is of great significance for optimizing the tooth profile design of a cycloid gear and the tolerance allocation during the installation of a gear pair. It can also provide the error excitation data for the accurate dynamic analysis of cycloid drives. The cycloid-pin gear reducer was initially introduced by Botsiber and Kingston [1] and Pollitt [2]. Then a series of novel cycloidal reducers was presented to obtain higher rigidity [3; 4; 5] or more compact structure [6; 7; 8]. A typical structure of cycloid-pin gear reducer is shown in Figure 1, which includes a gearbox casing, an input crankshaft, an output mechanism, two cycloid gears, and pins. During the transmission process, the cycloid gear not only rotates counterclockwise with the input crankshaft around the center of the pin center circle, but it also rotates clockwise around its own center to drive the output mechanism to rotate. For each cycloid gear, ideally, all the pins are in contact with it during the transmission process, and half of the pins transmit the load. Due to the multi-tooth meshing characteristics of cycloid-pin drives, these drives are more sensitive to MEs and AEs, and the tooth interference can easily occur in the transmission process. However, MEs and AEs are inevitable in practice, and these errors will change the number of contact teeth, the load distribution between teeth, the torsional stiffness, and other meshing characteristics, thereby adversely affecting the TE and backlash of cycloid-pin drives. In the published works, [PERSON] and [PERSON] [9; 10] first studied the influence of manufacturing tolerances on torque ripples and backlash, but only the profile tolerance was considered in their analysis model. [PERSON] et al. [11] investigated the influence of the manufacturing tolerance on the TE of a cycloidal gear reducer, and more types of manufacturing tolerance were considered in their analysis model, such as the pin radius error, pin position error, tooth profile error, and pitch error in the tooth profile. [PERSON] et al. [12] also studied the effects of tooth profile and pitch errors on the TE and backlash of a cycloidal-pin gear, achieving the effective pre-control of the tooth profile and the meshing contact performance. [PERSON] et al. [13] investigated the influence of the clearance and eccentricity errors on the gear ratio, transmission error, bearing load, and contact stress of a cycloid reducer. [PERSON] and [PERSON] [14] analyzed the global sensitivity of the transmission accuracy for cycloid-pin drives with the Sobol method in comprehensive consideration of the nonlinear and uncertainty factors, such as the MEs, AEs, and bearing clearances. [PERSON] et al. [15] studied the effects of the profile geometrical parameters of a trochoidal gear on the gearing process, clearance height change, and pulsation of a drive moment. [PERSON] et al. [16; 17] analyzed the TE of K-H-V-type planetary gears with cycloid gears based on a two-dimensional model with twenty degrees of freedom. [PERSON] et al. [18] analyzed the effects of the kinematics parameters on the transmission performance of a 2K-H pin-cycloid planetary mechanism based on a static analysis of the actual meshing transmission process between the gear pairs considering friction.