## 1 Introduction  
A counter-rotating output transmission system which adopts a power shunt when transmitting power can meet the requirements of high-speed and heavy-duty working conditions with a small volume and mass, and this system is widely used in helicopters, torpedoes, underwater unmanned boats, and aviation. For redundancy and safety, multiple inputs and multiple-branch shunt-convergence drives are generally used. However, due to the inevitable manufacturing and installation errors of the gear system, as well as the influence of factors such as the deformation of the moving components, the gap size between each branch and the output gear is inconsistent, resulting in an uneven distribution of the load transmitted by each branch, which seriously affects the power density and reliability of the transmission system. The size of the torsional stiffness of each branch is also an important factor affecting the load distribution. Therefore, it is important to study the matching relationship between the clearance and the torsional stiffness of each branch gear and optimize the corresponding parameters to reduce the unevenness of the load distribution of each branch and improve the reliability of the transmission system. To determine how to allocate the side clearance of the gearbox with the stiffness of each branch to obtain a reasonable load distribution coefficient, scholars around the world have performed a large amount of research in this area. (1) Calculation of load distribution based on hydrostatic gear split-torsion transmission  
[PERSON] et al. [1; 2] conducted a theoretical analysis and experimental study on the load distribution of a gear split-torsion transmission system under a static load using a synchronous angular load equalization design method without any load equalization device. [PERSON] and [PERSON] [3; 4] conducted a hydrostatic study on the equal load problem of multi-branch power-shunt reducers and derived equal load coefficients satisfying the prevailing manufacturing and installation conditions. [PERSON] et al. [5; 6] investigated the static mean load characteristics of gear shunt drive wheel systems with different structures based on deformation coordination conditions using the concentrated parameter theory. [PERSON] and [PERSON] [7; 8; 9] investigated the effects of the error, torsional stiffness, and standoff stiffness on the static load matching performance of multi-branch gearing systems. (2) Optimization of the load-equalization performance of a gear split-torsion transmission system based on a flexible shaft device  
[PERSON] et al. [10] proposed the use of elastic devices, such as elastic torsion shafts and rectangular elastic pads, to optimize the load distribution in gear shunt transmission systems. [PERSON] [11] proposed the design method of nesting a flexible shaft inside a duplex shaft, which resulted in a better load equalization performance of the system. [PERSON] et al. [12; 13] used a flexible shaft device in a multi-branch gear shunt transmission system and conducted a static and dynamic experimental study on the load equalization performance of the multi-branch transmission system. [PERSON] et al. [14] studied the equal load characteristics of the elastic torsion shaft system for a dual-input gear split transmission system and proposed a calculation method for the torsional stiffness of the elastic torsion shaft and the transmission error of the system. [PERSON] [15] studied the equal load characteristics of a dual-branch gear transmission system and proposed optimization measures for the flexibility of the wheel spokes and the support shaft. (3) Analysis of gearbox backlash based on three-dimensional tolerance  
The three-dimensional tolerance analysis model of Jacobian-torsor takes the backlash at the output end of the gearbox as the demand dimension and the other errors affecting the backlash as the constituent ring dimensions, which are calculated with the matrix method to avoid the complex derivation of geometric relationships. [PERSON] et al.