  This work links optimization approaches from hierarchical least-squares
programming to instantaneous prioritized whole-body robot control. Concretely,
we formulate the hierarchical Newton's method which solves prioritized
non-linear least-squares problems in a numerically stable fashion even in the
presence of kinematic and algorithmic singularities of the approximated
kinematic constraints. These results are then transferred to control problems
which exhibit the additional variability of time. This is necessary in order to
formulate acceleration based controllers and to incorporate the second order
dynamics. However, we show that the Newton's method without complicated
adaptations is not appropriate in the acceleration domain. We therefore
formulate a velocity based controller which exhibits second order proportional
derivative convergence characteristics. Our developments are verified in toy
robot control scenarios as well as in complex robot experiments which stress
the importance of prioritized control and its singularity resolution.
