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This paper presents methodologies for ensuring forward invariance of sublevelsets of constraint functions with high-relative-degree with respect to thesystem dynamics and in the presence of input constraints. We show that suchconstraint functions can be converted into special Zeroing Control BarrierFunctions (ZCBFs), which, by construction, generate sufficient conditions forrendering the state always inside a sublevel set of the constraint function inthe 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. Weconclude with a comparison of system trajectories under the two ZCBFs developedand prior literature, and a case study for an asteroid observation problemusing quadratic-program based controllers to enforce the ZCBF condition.

High Relative Degree Control Barrier Functions Under Input Constraints