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This paper presents the application of operational space control based on hierarchical task optimization for quadrupedal locomotion. We show how the behavior of a complex robotic machine can be described by a simple set of least squares problems with different priorities for motion, torque, and force optimization. Using projected dynamics of floating base systems with multiple contact points, the optimization dimensionality can be reduced or decoupled such that the formulation is purely based on the inversion of kinematic system properties. The present controller is extensively tested in various experiments using the fully torque controllable quadrupedal robot StarlETH. The load distribution is optimized for static walking gaits to improve contact stability and/or actuator efficiency under various terrain conditions. This is augmented with simultaneous joint position and torque limitations as well as with an interpolation method to ensure smooth contact transitions. The same control structure is further used to stabilize dynamic trotting gaits under significant external disturbances such as uneven ground or pushes. To the best of our knowledge, this work is the first documentation of static and dynamic locomotion with pure task-space inverse dynamics (no joint position feedback) control.