With the deepening of space exploration missions， space manipulators have become the core execution mechanisms for strategic tasks such as on-orbit servicing and deep space exploration. However， vibrations during launch， temperature fluctuations in a microgravity environment in space， and component aging caused by cosmic radiation can lead to dynamic drift in the hand-eye relationship of on-orbit manipulators， thereby reducing performance and increasing mission risk. To address these issues， this paper proposes an efficient and high-precision online hand-eye calibration approach suitable for the limited satellite computing environment. First， this paper derives a linear equation system for simultaneously solving the rotation and translation components of the hand-eye relationship， transforming the AX=XB problem into a one-step calculation， and derives the least squares solution. Subsequently， utilizing configuration from the manipulator controller（i.e.，A） and visual measurement data from the camera （i.e.，B） acquired during the execution of routine tasks by the on-orbit manipulator， the forgetting factor recursive least squares （FFRLS） algorithm is employed to achieve online hand-eye calibration. Finally， comparative experiments are conducted using a dataset constructed to simulate the on-orbit dynamic changes of the hand-eye relationship. Results show that， under the same computing power conditions， the proposed method requires only 0.75% of the computation time of the Tsai method combined with a sliding window， while achieving comparable calibration accuracy and demonstrating stronger robustness to measurement noise.