## 2 _Introduction_  
As the weakest component in bearings, cylindrical rollers bear most of the bearing load. Over 30% of bearing failures are caused by fatigue damage to the rolling elements [1]. Traditional methods for large-scale precision processing of cylindrical rollers mainly include centerless grinding [2; 3; 4; 5] and ultra-precision lapping [6; 7]. During these processes, the lapping tools are in a constant state of wear, which can affect the size consistency of rollers produced in batches [4; 8; 9]. In contrast, the double-disk lapping method for bearing balls can achieve a size and consistency precision that is an order of magnitude higher than rollers [10; 11; 12; 13; 14; 15]. Our research group has proposed a Double-Disk Straight Groove Lapping (DDSGL) method for the rolling surface processing of cylindrical rollers based on the batch lapping method for bearing balls. This processing, which shares a similar layout with the bearing ball processing, includes processing areas and external circulation devices and adopts a multi-cycle lapping processing method. Currently, the cylindrical rollers processed by DDSGL can achieve the highest grade G1 of ISO [16; 17]. In recent years, many scholars have been dedicated to improving the size consistency of cylindrical rollers based on double-disk lapping, which is a key technical indicator affecting bearing life and performance. [PERSON] et al. [18; 19; 20] explored the roller trajectories in a double-disk configuration. Through judicious parameter selection, they improvedthe accuracy of roller size and shape, ensuring nearly uniform processing across the roller surface. [PERSON] et al. [16; 17] utilized an elastic floating mechanism, similar to that used in bearing ball processing, to load cylindrical rollers. This enabled simultaneous processing of multiple rollers in the disk, leading to a significant improvement in size consistency. [PERSON] et al. [21] investigated the self-rotation principle of rollers in DDSGL, facilitating improved roller motion during processing. [PERSON] et al. [22] conducted an in-depth study of the roller forming mechanism in DDSGL, obtaining the stability conditions for the forming process in DDSGL. A review of these studies reveals that current researchers primarily focus on the processing areas in the field of double-disk lapping of cylindrical rollers. There is a lack of research on control strategies related to external circulation devices, which are more often designed as auxiliary mechanisms. However, external circulation devices play a crucial role in DDSGL. The number of rollers processed within the double disk is limited, and a large number of rollers need to be temporarily stored in the external circulation devices. Meanwhile, the size convergence results from the size comparative processing of rollers within the grinder can be spread to the entire batch of rollers through circulation, achieving collective size consistency convergence for the entire batch of rollers. In the well-established bearing ball lapping process system, a relatively complete external circulation device exists (as shown in Figure 1), which mainly includes a distributor, ball collector, ball flow groove, ball return groove, and vertical ball mixer. Most of these mechanisms serve to ensure the stable entry and exit of bearing balls from the processing area, a function similar to that of the DDSGL external circulation device. However, the vertical ball mixer in the bearing ball circulation device provides a mixing function that the DDSGL circulation system lacks. The primary function of the vertical ball mixer is to stir and forcibly disrupt the order of the bearing balls throughout the entire circulation process, thereby allowing balls located far apart in the cycle to have the opportunity to enter the processing area simultaneously for size comparison processing.