  The ability of executing multiple tasks simultaneously is an important
feature of redundant robotic systems. As a matter of fact, complex behaviors
can often be obtained as a result of the execution of several tasks. Moreover,
in safety-critical applications, tasks designed to ensure the safety of the
robot and its surroundings have to be executed along with other nominal tasks.
In such cases, it is also important to prioritize the former over the latter.
In this paper, we formalize the definition of extended set-based tasks, i.e.,
tasks which can be executed by rendering subsets of the task space
asymptotically stable or forward invariant. We propose a mathematical
representation of such tasks that allows for the execution of more complex and
time-varying prioritized stacks of tasks using kinematic and dynamic robot
models alike. We present and analyze an optimization-based framework which is
computationally efficient, accounts for input bounds, and allows for the stable
execution of time-varying prioritized stacks of extended set-based tasks. The
proposed framework is validated using extensive simulations and experiments
with robotic manipulators.
