This paper deals with motion planning algorithms for the large space robot manipulators with complicated dynamic behavior. We propose two "two-stage" iterative algorithms, which provide collision-free robot motion taking into account robot's dynamics. The approach is based on new efficient methods for robot manipulator dynamics simulation and probabilistic methods for motion planning in highly cluttered environments. The algorithms are applicable for the robot manipulators of general class with arbitrary kinematics and dynamics parameters. We have demonstrated the approach for a particular task of servicing the satellite by a large space manipulator. This task is one of the most challenging since large space manipulators have extremely complicated dynamic behavior caused by elasticity of their structure, huge payloads they work with and zero-gravity conditions. Experiments involving a 15.5 meters long manipulator carrying a satellite inside a space shuttle with clearance less than 3 cm are presented. Several movies demonstrate the results.