## 1 Introduction  
With the rapid development of mobile robots and autonomous driving, visual simultaneous localization and mapping (SLAM) has become a key technology. It allows these systems to perceive and localize themselves in unknown environments [1, 2]. SLAM has made significant progress over the past decade. Classic SLAM algorithms based on point clouds [3, 4, 5, 6], meshes [7, 8], voxels [9, 10, 11], or truncated signed distance function [12, 13, 14] have been developed. Recently, 3D reconstruction technologies have advanced, particularly with methods like NeRF (Neural Radiance Field) [15, 16, 17] and 3 DGS (3D Gaussian splatting) [18, 19, 20]. This has led to breakthroughs in SLAM research. These technologies allow for more accurate and realistic 3D reconstructions. They also improve localization and mapping using images or depth data. As a result, new SLAM systems based on 3D reconstruction technologies have emerged. NeRF has been successfully applied to SLAM systems, demonstrating the profound impact of new 3D reconstruction methods on SLAM technology. NeRF excels in view synthesis, detailed 3D reconstruction, and the ability to handle complex lighting and material effects [15]. Many NeRF-based SLAM frameworks leverage its strengths in generating fine-grained 3D scene representations [21; 22; 23], significantly improving localization and mapping accuracy in various scenes, and achieving more stable results. However, the use of ray-based volumetric rendering techniques to generate high-resolution rendered images significantly limits NeRF's rendering speed. Additionally, the implicit structure of NeRF, based on multilayer perceptron (MLP), lacks intuitiveness, which poses certain challenges for tasks such as scene understanding and editing [24]. Unlike NeRF, 3 DGS uses differentiable volumetric representations that are unstructured and explicit [18]. Instead of ray-based volumetric rendering, 3 DGS utilizes a projection-based approach to rasterization. This structure not only enables faster high-resolution rendering but also, due to its explicitness, makes scene editing and fine-tuning more convenient. Furthermore, 3 DGS ensures a high rendering quality while achieving more efficient computation and greater flexibility, especially in applications that require rapid iteration and optimization. These advantages make 3 DGS a better fit for SLAM systems, offering stable and efficient localization and mapping performance. Building on these advantages, many SLAM systems based on 3 DGS have been proposed [25; 26; 27; 28]. These systems leverage the strengths of 3 DGS to design and optimize SLAM from different perspectives, achieving high-precision localization and rendering. However, in the 3 DGS-based tracking process, the incompleteness of the scene map can lead to errors in the pose estimation of keyframes. Densifying the map with these erroneous poses may adversely affect its quality. To address this issue, we propose a novel SLAM system based on 3D Gaussians. In this paper, we propose a novel SLAM system based on 3D Gaussians, which is called Related Keyframe Optimization Gaussian SLAM (RK-SLAM). In this system, the latest keyframe is not directly used to densify the existing Gaussian map. Instead, a sub-Gaussian is established and associated with the pose of the keyframe. This sub-Gaussian undergoes refinement following the keyframe pose. Only after optimization is the refined sub-Gaussian used to densify the existing scene map. Moreover, for better tracking and mapping, we design a related keyframe selection algorithm. The algorithm selects keyframes for optimization together through inter-frame co-visibility and spatial distance between frames to improve the performance of tracking and mapping. In summary, our contributions are as follows:  
1. We propose an innovative SLAM system that leverages sub-Gaussian to refine the pose of keyframes and their associated Gaussian. This approach mitigates the impact of keyframe pose errors on densifying the existing map, ensuring a more accurate and robust mapping. 2.