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The basis for all advanced manipulator control is a relationship between the cartesian coordinates of the end-effector and the manipulator joint coordinates. A direct method for assigning link coordinate systems and obtaining the end effector position, and Jacobian, in terms of joint coordinates is reviewed. Techniques for obtaining the solution to these equations for kinematically simple manipulators, which includes all commercially available manipulators, is presented. Finally the inverse Jacobian is developed from the solution.
Paul et al. (Sun,) studied this question.