## 1 Introduction
[16] proposed a three-dimensional tolerance analysis model based on small displacement torsor (SDT) theory and a Jacobian matrix for characterizing the dimensional and form tolerances in geometries. In [PERSON] et al. [17], three-dimensional tolerance analysis methods in product design were studied. In [PERSON] [18], a dummy functional unit approach was proposed to simplify parallel chains by integrating tolerance information into the torque parameter of a specified dummy functional unit. [PERSON] et al. [19] developed a Jacobian-torsor volume tolerance analysis model based on the interval algorithm, which could perform tolerance analysis based on the extreme value method and statistical methods. [PERSON] [20] developed an error propagation model based on the Jacobian-torsor theory. [PERSON] et al. [21; 22; 23] proposed a contact analysis method for the rapid determination of the gear-drive clearance based on the Jacobian-torsor model and the Monte Carlo simulation technique. The existence of local parallel chains in the process of gear backlash analysis affects the accuracy of backlash analysis. [PERSON] et al. [24] compared the analytical results of spiral level gearboxes with and without partial parallel chains and proved that the effect of partial parallelism on the resolution accuracy was not negligible. The authors of [25; 26; 27; 28; 29] studied linear construction and parallel construction methods and proposed a probabilistic approach using a connected assembly model without considering partial parallel effects. [PERSON] et al. [30] studied a method to solve the partial parallel chain problem caused by deterministic deviations and established a generalized deviation propagation formulation for n-level component assembly. [PERSON] et al. [31; 32; 33] proposed the spin measure model to transform parallel chains into serial chain tolerances with complex algebraic operations. [PERSON] [34] proposed the use of the geometric structure leverage effect and combined the advantages of the CLIC (an acronym for \"localization tolerancing with contact influence\") method and the analytical line method to generate new serial chain tolerances. The current methods for calculating the load distribution based on static and elastic shaft structures do not take into account the effect of tolerance zones on the gear backlash distribution or three-dimensional tolerance analysis for parallel shafts, and there has been less research on multi-branch split-torsion transmission. The dual-input counter-rotating drive system has a complex structure and many factors affect the load distribution. In this research, we consider the effects of the backlash and elastic shaft torsional stiffness on the equal load performance at the output of two branches of the dual-input-to-rotation transmission system, and we establish a three-dimensional tolerance Monte Carlo analysis model of gear backlash based on the small displacement torsor theory and Jacobian matrix. The method of local parallel chain theory of gearbox meshing tooth pairs is proposed, and the calculation method of an elastic deflection angle and an equal load coefficient of the gearbox is combined with the theory of the elastic torsional deformation of the shaft system. The side clearance and the range of the equal load coefficient are calculated for the dual-input counter-rotating gearbox, and they are then verified through tests and proposed measures to optimize the equal load performance of the system.