After a brief summary of the basic aspects of displacement-group theory, several applications of this theory to the kinematics of robot mechanisms are described. Then, we show in greater detail how to obtain, in a simple way, the characteristic polynomials of manipulators with special structural parameters. The goal is to find the polynomials of degree two for these special-geometry manipulators. In our paper, we apply displacement-group theory to provide an elegant and compact presentation and discussion of this subject, and we demonstrate that some geometrical conditions on the shapes of the manipulator links, as stated in other scientific papers, can be eliminated without changing the degree of the characteristic polynomials.

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