Abstract
This is a mathematical study on the properties and the characteristics of the quadra-harmonic mean type manipulability index (H2).
There are several essential properties of manipulability indices which ensure that an index can be conveniently applied to actual theoretical or numerical analyses of robotic arms; i.e. that the index is: (1) scalar valued; (2) based on the singular values of Jacobian matrix; (3) independent of the task space coordinates; (4) expressed as an analytic function; and that (5) the index has a closed algorithm. But few studies have paid enough attention to them.
This paper discusses first the geometric meanings of the index H2, and it shows that the inverse of the index value coincides with the square mean root velocity in the joint coordinates space corresponding to the randomly given reference velocity in the task space.
Second, it is proved that the index H2 has the essential properties described above. Special attention is paid to the analyticity of the index function.
Furthermore, it is also proved that the value of the index H2 approximates minimal singular value of the Jacobian matrix within a neighborhood of each kinematic singular point, while it approximates the arithmetic mean of singular values within a neighborhood of each kinematically isotropic point. These approximated values express the manipulability best, but they do not have closed formulae nor simple numerical algorithms. These approximations are also illustrated graphically.
