SINGULARITY CONDITION OF WRIST-PARTITIONED 6-R SERIAL MANIPULATOR BASED ON GRASSMANN-CAYLEY ALGEBRA

Luc Djimon Clément Akonde

Abstract: To prevent the singularity of serial robot’s due to the lost of one or more degree of twist freedom, it is necessary to determine the Jacobian matrix J associated to its instantaneous motion and analyze the vanishing condition of the determinant det (J). Usually, the large expression of det (J) does not facilitate an efficient geometric analysis. Since Grassmann-Cayley Algebra (GCA) has powerful tools for geometric interpretation of coordinate free representation and singularity analyzing in real time computing, this method is implemented in the present work. The goal of this research is to determine the singularity condition of wristpartitioned 6-R serial manipulator (SM) based on GCA. The symbolic approach of Plücker coordinate lines is used to formulate the twist system (TS) of SM. The twist system is similar to det (J) which rows are Plücker coordinate lines. The vanishing condition of det (J) based on the linearity condition of TS is determined without algebraic coordinate and provides a single singularity condition which contains all generals and particulars cases. The keys elements of transition between the rows of J and singularity condition of a twist-partitioned 6-R SM are the introduction of the symbolic approach of Plücker coordinate lines and superbracket. The vanishing points of the superbracket are analyzed to describe the singularity condition. The result indicates that for a wrist-partitioned 6-R SM a single singularity condition contains three generals cases such as the shoulder, elbow and wrist singularity. Since the three last axis of the wrist are no-coplanar and intersecting at a unique point, it is suggested that for a wrist-partitioned 6-R serial manipulator, the wrist singularity never occurs physically, except that its design will be modified.

Keywords: Singularity, 6-R Serial Manipulators, Grassmann-Cayley Algebra, Projective space, Twist graph

DOI: https://doi.org/10.15623/ijret.2013.0212119