## 1 Introduction  
With rapid advancements in the industrial landscape and innovations in science and technology, increasing emphasis is being placed on achieving customized production through human-robot collaboration and on establishing sustainable manufacturing systems. These shifting demands are significantly influencing the design and performance of both industrial and service robots, particularly highlighting the importance of high-performance reducers to meet the precision and flexibility required in modern manufacturing environments. As user demands for high-performance reducers diversify, tooth profile design is gaining attention as a critical factor influencing reducer efficiency and performance. Representative tooth profiles include the cycloidal and involute profiles. The cycloidal profile, characterized by isochronism and the brachistochrone curve, contributes to stable power transmission and reduced energy loss, enabling high efficiency and low friction. The involute profile has revolutionized gear design with its ability to maintain a constant velocity ratio and ease of manufacturing. Ongoing research, supported by technological advances and innovation in related fields such as motors and controllers, continues to enhance the performance of robotic reducer systems. The cycloidal tooth profile offers a high reduction ratio and durability and is characterized by excellent torque transmission efficiency and low backlash. Its structure, in which multiple teeth engage simultaneously, averages out errors and is widely evaluated as suitable for high-precision reducers. Based on these advantages, theoretical, technical, and applied research is actively being conducted. [PERSON], [PERSON], and [PERSON] proposed a new trapezoidal tooth profile reducer for robot manipulators, aiming to simplify the design and optimize performance through differentiation from conventional profiles. They validated the effectiveness of the proposed profile through mechanical analysis and testing [1]. [PERSON] et al. studied tooth profile modification to improve the performance of cycloidal reducers, analyzing the effects on load capacity, noise reduction, vibration mitigation, and transmission accuracy. Their study demonstrates that tooth profile modification is a key factor in optimizing reducer design and performance [2]. [PERSON] et al. analyzed the design and dynamic characteristics of a modified cycloidal reducer applying an epitrochoidal tooth profile, demonstrating that the new profile improves efficiency and stability [3]. [PERSON] et al. proposed a new method for modifying cycloidal gear tooth profiles used in precision reducers for robots. By developing a mathematical model that considers pressure angle distribution and backlash, they enhanced meshing characteristics and power transmission performance, thereby improving design flexibility and transmission accuracy [4]. [PERSON] et al. designed and manufactured an internal noncircular gear based on an improved cycloidal profile to overcome the limitations of conventional profiles and optimize motion efficiency and gear performance [5]. Overall, reducers using cycloidal tooth profiles offer excellent characteristics, including compact design, low backlash, high accuracy, and high efficiency [3; 6]. Due to its structural characteristics, the harmonic drive--initially called the \"strain wave gearing system\"--comprises a wave generator, a flexspline, and a circular spline. It is known for its high reduction ratio, lightweight design, and high efficiency [7; 8; 9; 10]. These advantages have led to its widespread adoption in aerospace applications such as solar panel deployment mechanisms, antenna pointing systems, and planetary rover wheel drives, as well as in industrial robots [11; 12]. To improve the performance of harmonic drives, extensive research on tooth profile design is ongoing.