The paper presents algorithms of automatic interactions of the robot manipulator with moving objects - one or two tennis balls, suspended on the threads, which perform unknown motion under the action of natural forces. The interaction algorithms are based on the use of vision system and the models of the objects dynamics with unknown parameters, defined by the initial conditions. The values of these parameters are calculated as the result of the analysis of the objects positions, defined by vision system. Further, motion of the balls is predicted and robot motion is scheduled. The experiments were focused on modifying the motion of the ball in such a way for its orbit got the desirable parameters. In particular, the ball was directed to reach the motionless target placed inside the working space, or perform a rendezvous with another ball having the independent free motion, and, at last, to provide a rendezvous with another ball by such a way, that the second ball reaches the motionless target. The operations are enumerated from simplest to the most difficult one and they are performed with various reliability. Experimental estimations of the reliability allowed considering limit possibilities of the control system. The paper presents characteristics of experimental setup and the results of the experiments.