# Traceability and Spectral Analysis of Transmission Error in Rotary Vector Planetary Gear Reducers
## 1 Introduction  
With the rapid development of modern industry, the demand for high-precision reducers in humanoid, collaborative, and service robots has significantly increased [1, 2]. The rotary vector (RV) reducer, derived from cycloidal pinwheel (CYCP) transmissions, integrates cycloidal and planetary gear mechanisms. It is valued for its compact design, ease of assembly, high precision, and long service life. These attributes make RV reducers widely used in high-precision applications, including industrial robots, CNC machines, military equipment, and medical devices [3, 4]. However, inherent errors caused by manufacturing, assembly, load deformation, and thermal effects in gear transmissions make transmission error (TE) unavoidable [5, 6]. These errors not only reduce transmission accuracy and motion stability but also induce vibrations and shocks under complex working conditions, ultimately shortening the reducer's service life [7, 8]. Scholars have shown significant interest in the transmission accuracy and dynamic characteristics of RV reducers, resulting in various analytical models. Some studies have focused on optimizing cycloidal gear design. For instance, [PERSON] et al. [9] proposed a segmented improvement method for cycloidal gears, achieving higher transmission accuracy than traditional correction methods. [PERSON] et al. [10] identified clearance in CYCP components and crankshaft installation errors as key sources of backlash, with simulations and experiments validating their mathematical model. [PERSON] et al. [11] developed a CYCP model to evaluate the impact of manufacturing errors in cycloidal gears on transmission accuracy. [PERSON] et al. [12], drawing on studies of spur and helical gears, derived an optimaltooth modification formula to minimize TE. Using finite element analysis, [PERSON] et al. [13] explored the effects of tolerances and friction between the cycloidal disk and pins on RV reducer performance. [PERSON] et al. [14] proposed an elastic TE compensation method to identify factors improving transmission accuracy. [PERSON] et al. [15] introduced the concept of optimal measurement speed for precise TE calculation. Modeling, analyzing, and optimizing TE are key areas of research. [PERSON] et al. [16] proposed a dynamic TE analysis method for RV reducers, accounting for machining errors, assembly errors, and flexibility factors. [PERSON] et al. [17] developed a lumped-parameter dynamic model to study the effects of machining errors, assembly errors, and bearing clearance on transmission accuracy. [PERSON] et al. [18] used orthogonal simulation to evaluate the impact of various bearing clearances on diagonal TE. [PERSON] et al. [19] proposed a dynamic model for the CYCP mechanism in RV reducers, analyzing transmission characteristics and dynamic contact responses. [PERSON] et al. [20] introduced a multi-teeth contact model and TE model, investigating how load influences different modification methods. [PERSON] et al. [21] developed an error analysis model using an equivalent mechanism to explore the impact of initial errors on transmission accuracy in both high- and low-speed stages. Recent advancements in artificial intelligence methods have highlighted virtual prototyping and TE prediction as key research areas for RV reducers. [PERSON] et al. [22] used virtual prototyping to analyze the primary factors influencing dynamic TE. [PERSON] et al. [23] applied this technology to study transmission efficiency, gear forces, and TE. [PERSON] et al. [24] addressed uncertainties in manufacturing and assembly errors to predict the reliability of RV reducers. [PERSON] et al. [25] simulated involute gear profile errors to effectively predict transmission accuracy in drive systems. [PERSON] et al. [26] developed a high-precision regression model for rotational error prediction using optimal Latin hypercube sampling and a Deep Gaussian Process model.