Comments on Manipulability Measure in Redundant Planar Manipulators

In this paper we perform an analysis of the manipulability matrix of a manipulator in terms of its eigenvalues and eigenvectors (eigen-analysis), which defines the well know manipulability ellipsoid for planar manipulators. We show that the manipulability measure does not depend on the first joint angle, for redundant manipulators. The determinant of manipulability matrix doesn´t change when the first angle varies. So, as we´ll show, the product of the eigenvalues remains the same. The manipulability ellipsoid changes with the first joint angle, but keeps constant the manipulability measure (area of the ellipsoid). We claim that manipulability-control based algorithms must use the eigenvectors and eigenvalues of manipulability matrix independently, in order to be optimal. Some tests show the improvement of the control law when we use directly the eigenvectors as a local basis for the control. Furthermore we suggests that the control analysis should be done not only in the joint space, buy in the manifold spanned by the Manipulability matrix M, that should lead to naturally simple control laws that uses the optimal freedom in the joint space