One of the most important and widely used concepts in the kinematic analysis of robot manipulators is the reciprocal screw. However, there are no general expressions and easy methods to obtain the reciprocal screw in an analytic manner. This paper suggests an analytic formulation of the reciprocal screws of arbitrarily aligned screw systems. Since the reciprocal screws obtained in this paper are represented by the direction vectors and the position vectors of the given screws, we can analyze the relation between the reciprocal screw system and the given screw system easily. With the results, to find a reciprocal screw is to solve an algebraic equation of the corresponding system of screws. In order to show the usefulness of the result, several examples related to the robot manipulator are provided. For a nonredundant serial manipulator, the pseudo inverse of the Jacobian matrix is shown to be equivalent to the wrench matrix obtained by the reciprocity. For a parallel manipulator, a leg is isolated to obtain an independent part from the manipulator and is analyzed analytically. The proposed method can be applied to any arbitrarily aligned screw system.

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