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Visually guided robotic capturing of a moving object often requires long-term prediction of the object motion not only for a smooth capture but because visual feedback may not be continually available, e.g., due to vision obstruction by the robotic arm, as well. This paper presents a combined prediction and motion-planning scheme for robotic capturing of a drifting and tumbling object with unknown dynamics using visual feedback. A Kalman filter estimates the states and a set of dynamics parameters of the object needed for long-term prediction of the motion from noisy measurements of a vision system. Subsequently, the estimated states, parameters, and predicted motion trajectories are used to plan the trajectory of the robot's end-effector to intercept a grapple fixture on the object with zero relative velocity (to avoid impact) in an optimal way. The optimal trajectory minimizes a cost function, which is a weighted linear sum of travel time, distance, cosine of a line-of-sight angle (object alignment for robotic grasping), and a penalty function acting as a constraint on acceleration magnitude. Experiments are presented to demonstrate the robot-motion planning scheme for autonomous grasping of a tumbling satellite. Two robotics manipulators are employed: One arm drifts and tumbles the mockup of a satellite, and the other arm that is equipped with a robotic hand tries to capture a grapple fixture on the satellite using the visual guidance system.