## 1 Introduction
In the above studies, the analysis models of the TE and backlash were mostly established in a two-dimensional plane, and the MEs that were considered were mostly along the tooth profile direction, while the MEs along the tooth-width direction were rarely involved. Additionally, the studies of the effects of the AEs on the TE and backlash were incomplete. Only the influence of the eccentricity errors was analyzed, while the effects of misalignment deviations were not considered. To avoid the tooth interference caused by MEs and AEs and obtain better lubrication conditions, profile modifications (PMs) are necessary for cycloid-pin drives. This also has a non-negligible effect on the TE and backlash of cycloid-pin drives. [PERSON] et al. [5] proposed a compound modification method combining negative isometric and negative offset modification to reduce the influence of PMs on backlash. [PERSON] et al. [19] proposed a novel method for PMs to improve the motion accuracy and modification quality that comprehensively  
Figure 1: Typical structure of cycloid-pin gear reducer. considered the effects of the backlash, pressure angle distribution, and tooth tip and root clearance. In addition, many scholars have studied the torsional stiffness, transmission efficiency, load analysis, and optimization design methods of cycloid-pin drives. In terms of torsional stiffness, [PERSON] et al. [20] proposed a new method for estimating the elastic torsional compliance of single-stage cycloidal drives based on static experimental results. [PERSON] et al. [21] proposed a new method to calculate the mesh stiffness of a cycloid-pin gear pair considering the effects of PMs and the eccentricity error based on the unloaded tooth contact analysis and the nonlinear Hertzian contact theory. [PERSON] et al. [22] calculated the torsional stiffness of a cycloid gear using the Majumdar-Bhushan contact model and the finite element method. In terms of transmission efficiency, [PERSON] et al. [23] proposed a simplified procedure to calculate the force distribution on cycloid drive elements, the power losses, and the theoretical mechanical efficiency. [PERSON] et al. [24] analyzed the effects of design parameters on the efficiency of a cycloidal gear drive, such as the pitch radius and number of housing rollers. [PERSON] [25] derived the efficiency equations for cycloid drives using [PERSON]'s framework. In terms of load analysis, [PERSON] and [PERSON] [26] investigated the influence of the design parameters of a cycloid speed reducer on forces and contact stress. [PERSON] et al. [27] proposed a method for analyzing the contact dynamics of the multi-tooth meshing in a cycloid-pin drive considering the effect of the turning-arm cylindrical roller bearing. In terms of the optimization design methods of cycloid-pin drives, [PERSON] et al. [28] proposed an optimization design method for cycloid drives with the goal of maximizing efficiency and minimizing volume based on a genetic algorithm. [PERSON] et al. [29] proposed a multi-objective optimal model for cycloid-pin drives in which the objective functions were used for the bending stress of the pins, the force of the crank bearing, and the reducer volume. With the development of computing technology in recent years, the finite element method has been widely used in the research of cycloid-pin drives. For example, [PERSON] [30] carried out loaded gear contact analysis and strength calculations for trochoidal gear reducers based on a mechanics model and a finite element method. [PERSON] et al. [31] studied the lost motion of a cycloid reducer using a combined finite element and kinematic analysis and concluded that the backlash of a cycloid reducer was not only related to the torsional stiffness but also to the tolerance. However, the multi-tooth meshing characteristics of cycloid-pin drives often lead to a huge finite element model, low computational efficiency, and expensive computational costs. Cycloid-pin gear pairs are the key components of the cycloid-pin drives.