## 1 Introduction  
The equilibrium of tension between strands in cable twisting is of great significance to the improvement of the mechanical properties of cables [1, 2]. The cable-twisting production scheme is shown in Figure 1. The model of the cable-twisting equipment was established in the solidworks. During the stranding process, the tension of each strand may deviate due to different twisting conditions, and the strands may be squeezed out during twisting, resulting in broken or disordered strands. This may further lead to quality issues in the product and may result in more severe safety incidents, such as a cable-bearing fracture [3, 4].  
The early research work on cable-system mechanics mainly focused on the vibration mechanics analysis of the cable system and the mechanical performance analysis of the geometric structure [5, 6]. [PERSON] et al. [7] established a multidirectional coupling vibration model of the cable strand. They used the Galerkin truncation method to solve the model parameters and verified the strand dynamics model based on the experimental system. [PERSON] et al. [8] established a dynamic model of the single-rope winding lifting system based on the Hamilton principle to obtain the relationship between the tension deviation between two cables and the longitudinal vibration. [PERSON] et al. [9] introduced a displacement interpolation function based on the absolute node co-ordinate method to establish the structural deformation model of the strand under the axial load. [PERSON] et al. [10] established a spiral line equation with the periodic boundary condition based on the spatial analytical geometry method to obtain the relationship between the bending, torsion force, and tension of the strand. Domestic and foreign researchers have also performed relevant work on the tension deviation between strands or cables. [PERSON] et al. [11] combined the traditional and coupled disturbance observer through estimating the Coulomb friction and compensating for the coupled disturbance; they used a nonlinear inverse step control scheme to reduce the tension deviation of two cables in the lifting system. [PERSON] et al. [12] established a distributed dynamic model and designed a parameter-adaptive strand-tension-balance robust boundary controller to achieve cable-tension-balance control and longitudinal vibration suppression of the lifting container. [PERSON] et al. [13] analyzed the redundancy characteristics of each actuator to reduce the tension deviation of each strand under the impact, and proposed a hybrid control scheme based on iterative learning control and a radial basis function neural network to improve the performance of the controller. [PERSON] et al. [14] combined a low-gain state observer and a nonlinear robust adaptive controller to address the cable tension imbalance problem in the double-rope winding lifting system, and verified that the algorithm has a superior performance in strand tension equilibrium control.  
However, there has been limited research on the analysis of the forces on each strand and the control of the tension deviation under the twisting working condition of the cable. We explored the causes of the tension deviation of strands under different twisting working conditions, and analyzed the forces of strands at the guide hole of the cylinder and the twisting deflection of the distribution reel. The tension data of strands under different traveling paths and twisting-deflection paths were obtained through simulation experiments in the laboratory. A discrete sliding mode control method was used to respond in real time to dynamic changes in the strand tension, and reduced the influence of the above interference factors on the strand tension. Tension equilibrium and fluctuation reduction were ultimately achieved between the strands by adjusting the control parameters.