3R Wrist Positioning - a Classical Problem and its Geometric Background

The

wrist centre positioning problem

(WP) peculiar to the motion produced by the first three joints of a general six revolute jointed (6R), wrist partitioned serial robot and the underlying geometry is reexamined. Conventionally a sequence of six rotational operations, alternately in terms of known geometric parameters and unknown joint angles, expresses the desired position. However the solution can be represented by four intersection points between a fourth order cyclid, and a circle. Properties of the curves of intersection of the cyclid with the absolute plane reveal why the univariate polynomial (UVP) is of fourth degree rather than eighth as indicated by the Bezout number. Simple cyclid geometry makes it convenient to investigate specific 3R positioning architectures and expose degenerate cases.