Let $J()$:$R^n{R}^p$ be the Jacobian of a manipulator $(p=3$ or

$.$ Assume that a task is modeled by an ellipsoid in the task space

with its principal axes of length ${}_1,$cdots,${}_p.$ Let $E_{}sub{R}^p$ be an

ellipsoid of size scaled by $,$ namely

$E_{}$:$=\{y$:$(\frac{y_1}{{}_1}{)}^2+$cdots$+(\frac{y_p}{{}_p}{)}^2\}$

Define a manipulability measure on $J()$ which takes into account

the task requirement as

${}_t()$:$=$max$\{$:$J()(B_1)sub{E}_{}\}.$

Characterize ${}_t()$ in terms of the singular values of $J()$ and lengths

of the principal axes, ${}_1,$dots,${}_p.$