  This paper presents methodologies for ensuring forward invariance of sublevel
sets of constraint functions with high-relative-degree with respect to the
system dynamics and in the presence of input constraints. We show that such
constraint functions can be converted into special Zeroing Control Barrier
Functions (ZCBFs), which, by construction, generate sufficient conditions for
rendering the state always inside a sublevel set of the constraint function in
the presence of input constraints. We present a general form for one such ZCBF,
as well as a special case applicable to a specific class of systems. We
conclude with a comparison of system trajectories under the two ZCBFs developed
and prior literature, and a case study for an asteroid observation problem
using quadratic-program based controllers to enforce the ZCBF condition.
