Equivalent and Dual Robotic Manipulators through Dual Transformations, Reciprocal Screws and Graph Theory

Duality transformations have effective impacts on simplifying analysis and synthesis steps of systems, due to their additively topological richness in unification and generalization of theories. Duality between statics and kinematics of mechanical systems in general, and robotic manipulators in particular, aided the discoveries of novel dual structures which possess superiority among their topological types. According to the previous hypothesis, this thesis addresses topological dualities in engineering systems as a concept, and the duality between different structures (geometrical wise) of robotic manipulators in vivid. The latter duality was found naturally leading to the commonly known reciprocity between actively coordinated systems provided by the theory of screws. The major contribution of this thesis is represented by generalizing the geometrical reciprocity problem of having a set of screws which each of its elements is reciprocal to all the elements of another set of screws, except one. Accordingly, this generalization breaks the confines of duality from existing only between special cases of serial and parallel manipulators, and extended its boundaries to combine a wide range of structures. Moreover, the geometrical meaning for Moore-Penrosians’ pseudo inverses of Jacobians was clarified naturally by means of linear algebra. The latter resulted in a new insight for duality in robotic systems especially in terms of the usage of reciprocity leading to equivalency. Keywords: Duality, Electrical Mechanical Analogs, Robotic Equivalents, Kinematics, Statics, Graph Theory, Screw Theory and Reciprocal Screws.