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In response to the costly and error-prone manual satellite tracking on the International Space Station (ISS), this paper presents a deep neural network (DNN)-based robotic visual servoing solution to the automated tracking operation. This innovative approach directly addresses the critical issue of motion decoupling, which poses a significant challenge in current image moment-based visual servoing. The proposed method uses DNNs to estimate the manipulator's pose, resulting in a significant reduction of coupling effects, which enhances control performance and increases tracking precision. Real-time experimental tests are carried out using a 6-DOF Denso manipulator equipped with an RGB camera and an object, mimicking the targeting pin. The test results demonstrate a 32.04% reduction in pose error and a 21.67% improvement in velocity precision compared to conventional methods. These findings demonstrate that the method has the potential to improve efficiency and accuracy significantly in satellite target tracking and capturing.