In this paper we discuss singular configurations of serial robot-manipulators with respect to their removability. A removable singularity is a singularity which can be removed from a motion of the end-effector by a small change of the motion. The most interesting situation appears for robot-manipulators with 5 degrees of freedom, because the case of 4 degrees of freedom is easy and singular configurations of robot-manipulators with 6 degrees of freedom are nonremovable. In the paper we give the complete list of all 5R robot-manipulators which have nonremovable singularities. The image of the singular set in the parameter space for such manipulators can be a plane, a quadric, a cylinder or an algebraic surface of degree 3 or 7. All of them are explicitly given.

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