# Prediction of Tooth Profile Deviation for WEDM Rigid Gears Based on ISSA-LSSVM
## 1 Introduction  
Harmonic gear reducers offer numerous advantages, such as a significant transmission ratio, robust load-carrying ability, and exceptional transmission precision. Consequently, they find widespread application in aerospace, robotics, the defense industry, and other high-end precision technology fields [1, 2]. Given the precision transmission characteristics of harmonic gear reducers, it becomes crucial to guarantee the accuracy and quality of each mechanism during the manufacturing process. This is necessary to guarantee that the transmission performance of the finished parts aligns with the design requirements. Being the core component of the harmonic gear reducer, the tooth profile deviation of the rigid gear significantly impacts the performance and assembly performance of the harmonic gear transmission system. Numerous researchers have conducted extensive studies on gear tooth deviation in different machining processes. [PERSON] et al. [3] explored the computation approach of tooth profile deviation in conventional turning, assessed the impact of turning tool rake angle, and proposed a method to improve tooth profile deviation through turning tool grinding. [PERSON] et al. [4] developed a comprehensive tooth profile deviation model using conjugate surface meshing theory, the Box-Behnken experimental design, and the artificial immune clone algorithm. The model optimizes and actively controls the internal gear power honing (IGPH) process parameters to obtain high gear geometry accuracy. [PERSON] et al. [5] established quantitative mapping models for hob geometry errors and gear geometric errors, revealing essential mapping rules and laying a theoretical foundation for achieving higher precision in roll-cutting gears. [PERSON] et al. [6] provided a forecast model for hobbing geargeometric errors. They used a modified PSO-BP algorithm to analyze the correlation between the hobbing process parameters and tooth shape deviation. [PERSON] et al. [7] analyzed the tooth profile deviation in the hobbing forming process base on the theory of gear meshing, and adjusted the hobbing process parameters for the deviation size. [PERSON] et al. [8] performed a study analyzing how wire-cutting parameters, including pulse width, open circuit voltage, wire tension, and pulse current, affect the deviation of the straight-toothed cylindrical gear tooth profile through orthogonal experiments. [PERSON] et al. [9] found that adjusting hobbing parameters can greatly impact the microgeometry deviation and average deviation of gears. By determining the optimal combination of parameters, gear precision can be significantly improved. [PERSON] et al. [10] presented a model to analyze the time-varying meshing stiffness of asymmetric gear pairs, which yields outcomes more similar to primitive meshing by taking into account the tooth shape deviations. They also highlight the significant effect of small shape deviations on gear meshing characteristics. [PERSON] et al. [11] presented a new method to solve the meshing stiffness and analyzed the variation in the stiffness and load distribution ratio with tooth profile deviation during gear meshing. [PERSON] et al. [12] introduced a mathematical method to investigate the variation in tooth profile deviation that occurs when cutting gears under various parameters using the same turning tool. Their findings indicated that reducing the helix angle or the number of teeth can result in a change in tooth profile deviation. The existing literature primarily focuses on the gear processing mechanism and uses theoretical modelling and simulation analysis to explore the causes of tooth profile errors. Traditional machining methods, such as hobbing and shaping, have been extensively studied. However, these methods often face limitations in flexibility and precision, especially when dealing with complex gear geometries. Recent advances in gear manufacturing have highlighted the potential of wire electrical discharge machining (WEDM) for improving gear quality and performance.